% CHANGES TO VOLUME 3 OF THE ART OF COMPUTER PROGRAMMING
%
% Copyright (C) 1998,1999,2000,2001,2002,2003,2004,
%                 2005,2006,2007,2008 by Donald E. Knuth
% This file may be freely copied provided that no modifications are made.
% All other rights are reserved.
%
% Three levels of changes to the books are distinguished here:
%
% "\bugonpage" introduces the correction of an error;
% "\amendpage" introduces new material for future editions;
% "\improvepage" introduces ameliorations of lesser importance.
%
% (Changes introduced by \improvepage do not appear in the hardcopy listing.)
%
% Also, "\planforpage" introduces some of the author's half-baked intentions.
%

% NOTE: TO PUT THE INDEX ON A SEPARATE PAGE, RUN THIS WITH THE COMMAND LINE
%   tex "\let\indexeject+ \input err3"

\newif\ifall % \alltrue means show the trivial items too
\relax % hook

\def\vertical{|}
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\catcode`|=\active
\let|\inref \input \jobname.ref
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\input taocpmac % use the format for TAOCP, with modifications below

\def\becomes{\ifmmode\ \hbox\fi{\manfnt y}\ } % wiggly arrow indicates a change

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  \medbreak\defaultpointsize
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   {\bf Page #2}\enspace #3
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  \medbreak\defaultpointsize
  \line{\kern-5pt{\bf Page #2}\enspace #3
   \leaders\hrule\hfill\ \eightrm\date#4.}
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\def\improvepage#1.#2 #3 (#4) {\ifall
  \medbreak\ninepoint
  \line{\kern-6pt{\sl Page #2\enspace #3\/}
   \leaders\hrule\hfill\ \eightrm\date#4.}
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  \medbreak\defaultpointsize
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\def\date#1.#2.#3.{% convert "yy.mm.dd." to "dd Mon 19yy"
  #3
  \ifcase#2\or Jan\or Feb\or Mar\or Apr\or May\or Jun\or
               Jul\or Aug\or Sep\or Oct\or Nov\or Dec\fi
  \ \ifnum #1<97 \hundred#1\else19#1\fi}
\def\hundred{20} % the "century" for dates before '97

\def\ex #1. [#2]{\ninepoint
  \textindent{\bf#1.}[{\it#2\/}]\kern6pt}
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  \textindent{\llap{\manfnt x}\bf#1.}[{\it#2\/}]\kern6pt}

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\def\refin#1 {\let|\inref \input #1.ref \let|\crossref}

\let\defaultpointsize=\tenpoint

%%%%%%%%%%%%%% opening remarks %%%%%%%%%%%%%%%%%%%%%%%%

\def\lhead{INTRODUCTION}
\let\rhead=\lhead
\titlepage
\volheadline{THE ART OF COMPUTER PROGRAMMING}
\bigskip
\volheadline{ERRATA TO VOLUME 3 (2nd edition)}
\bigskip

\noindent This document is a transcript of the notes that I have been making
in my personal copy of {\sl The Art of Computer Programming}, Volume~3 (second
edition) since it was first printed in 1998.

\ifall Four levels of updates\dash---``errors,'' ``amendments,'' ``plans,''
   and ``improvements''\dash---appear, indicated by four
\else Three levels of updates\dash---``errors,'' ``amendments,'' and
   ``plans''\dash---appear, indicated by three \fi
different typographic conventions:
\begingroup\def\hundred{17}

\bugonpage 0.666 line 1 (76.07.04)
Technical or typographical errors (aka bugs) are the most critical items, so
they are flagged with a `\thinspace{\manfnt x}\thinspace' preceding the page
number. The date on which I first was told about the bug is shown; this is the
effective date on which I paid the finder's fee. The necessary
corrections are indicated in
a straightforward way. If,~for example, the book says
`$n$' where it should have said `$n+1$', the change is shown thus:
\smallskip
$n$ \becomes $n+1$
\endchange

\amendpage 0.666 line 2 (89.07.14)
Amendments to the text appear in the same format as bugs, but without
the~`\thinspace{\manfnt x}\thinspace'. These are things I wish I had known
about or thought of when I wrote the original text, so I added them later.
The date is the date I drafted the new text.
\endchange

\def\hundred{19}

\planforpage 0.666 line 3 (17.11.20)
Plans for the future represent a third kind of item. In such notes I~sketched
my intentions about things that I wasn't ready to flesh out further when
I~wrote them down. You can identify these items because they're written in
slanted type, and preceded by a bunch of dots
`\hbox to 6em{\leaders\hbox to 5pt{\hss.\hss}\hfill}' leading to the date on
which I recorded the plan in my files.
\endchange

\improvepage 0.666 line 4 (38.01.10)
The fourth and final category\dash---indicated by page and line number in
smaller, slanted type\dash---consists of minor corrections or improvements
that most readers don't want to know about, because they are so trivial.
You wouldn't even
be seeing these items if you hadn't specifically chosen to print the complete
errata list in all its gory details.
Are you sure you wanted to do that?
\endchange

\endgroup
\ifall\else\medskip\ninepoint
My personal file of updates also includes a fourth category of items, not
shown in this list. They are miscellaneous minor corrections or improvements
that most readers don't want to know about, because they are so trivial.
If you really want to see all of the gory details,
you can download the full list from Internet webpage
$$\.{http://www-cs-faculty.stanford.edu/\char`\~knuth/taocp.html}$$
by selecting the ``long form'' of the errata.
\fi

\medskip
\tenpoint
My shelves at home are
bursting with preprints and reprints of significant research results that I
want to digest and summarize, where appropriate, in the ultimate edition of
Volume~3. I didn't do that in the second edition because I would surely have to do
it over again later: New results continue to pour forth at a great rate, and I
will have time to rewrite that volume only~once. Volumes 4 and~5 need to be
finished first. So I've put most of my effort so far into writing up
those parts of the total picture that seem to have converged to their
near-final form. It follows, somewhat paradoxically, that the updates in this
document are most current in the areas where there has been least activity.

On the other hand I do believe that the changes listed here bring Volume~3
completely up to date in two respects: (1)~All of the research problems in the
previous edition\dash---i.e., all exercises that were rated 46 and
above\dash---have received new ratings of 45 or less whenever I learned of a
solution; and in such cases, the answer now refers to that solution.
(2)~All of the historical information about pioneering developments has been
amended whenever new details have come to my attention.

\beginconstruction
The ultimate, glorious, future editions of Volumes 1--3 are works in progress.
Please let me know of any improvements that you think I ought to make.
Send your comments either by snail mail to D.~E. Knuth, Computer
Science, Gates Building 4B, Stanford University, Stanford CA~94305-9045,
or by email to
{\tt taocp{\char`\@}cs.stanford.edu}. (Use email for book suggestions
only, please\dash---all
other correspondence is returned unread to the sender, or discarded,
because I have no time to
read ordinary email.) Although I'm working full time on
Volume~4 these days, I~will try to reply to all such messages
within a year of receipt. Current news about {\sl The Art of Computer
Programming\/} is posted on
$$\.{http://www-cs-faculty.stanford.edu/\char`~knuth/taocp.html}$$
and updated regularly.
\par\endconstruction
\rightline{\dash---Don Knuth, February 1998}

\bigskip
\bigskip
{\quoteformat
Writing a series like {\rm The Art of Computer Programming}
is similar to painting the Forth Rail Bridge.
No sooner is it finished than
the job must be started again.
\author MALCOLM CLARK (1992)
% A plain TeX primer, Oxford University Press, page 7
\bigskip\bigskip
The time when\/ {\rm The Guardian} ceases to make mistakes altogether
is not, at the moment, foreseeable.
\author IAN MAYES (1998)
% quoted in IHT, 17 Feb 98, p5
% he is Reader's Editor of the Manchester Guardian
\vfill\eject
}

\def\today{\number\day\space\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\year}

%%%%%%%%%%%%%%% CHANGES FOR VOLUME 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\lhead{CHANGES TO VOLUME 3: SORTING AND SEARCHING}
\let\rhead=\lhead
\titlepage
\volheadline{SORTING AND SEARCHING}
\bigskip
\rightline{Copyright \copyright\ 1998, 1999, 2000, 2001, 2002, 2003, 2004,\qquad}
\rightline{2005, 2006, 2007, 2008, Addison\with Wesley; all rights reserved}
\rightline{Last updated \today}
\bigskip
\rightline{\sl Most of these corrections have already been made in
                  recent printings.}
\smallskip
\let\defaultpointsize=\tenpoint

\bugonpage 3.1 line 7 (98.05.05)
{\eightssi The Prince\/} \eightss (1951) \becomes
{\eightssi The Prince\/} \eightss (1513)
\endchange

\improvepage 3.5 line 3 of exercise 3 (98.11.16)
\ninepoint conditions (1) and (2) \becomes conditions \eq(1) and \eq(2)
\endchange

\bugonpage 3.6 line 1 of exercise 9 (02.08.24)
\ninepoint After $n$ \becomes After $N$
\endchange

\bugonpage 3.7 line 2 (00.01.03)
at most ten \becomes less than a dozen
\endchange

\improvepage 3.7 line $-24$ (98.09.14)
\ninepoint Achtzehnhundert zw\"olf \becomes Achtzehnhundertzw\"olf
\endchange

\improvepage 3.7 line $-7$ (98.11.11)
\ninepoint s\"ussen M\"adeln. \becomes langen Tag.
\endchange

\improvepage 3.8 line 31 (03.02.16)
\ninepoint upper case letter \becomes uppercase letter
\endchange

\improvepage 3.12 near the top (01.12.19)
line 9: Marshall Hall's \becomes the simple\nl
line 11: [See {\sl Proc.\ }\dots 203.] We \becomes We
\endchange

\improvepage 3.14 line 7 (01.10.09)
K. F. Hindenburg \becomes C. F. Hindenburg
\endchange

\bugonpage 3.15 line 2 after {\eq(7)} (00.10.28)
Rodriguez \becomes Rodrigues
\endchange

\bugonpage 3.17 line 3 (98.05.12)
$\bigl((a_1\,a_2\ldots a_n),\;(p_1,p_2,\ldots,p_n)\bigr)$ \becomes
$\bigl((a_1,a_2,\ldots,a_n),\;(p_1,p_2,\ldots,p_n)\bigr)$
\endchange

\bugonpage 3.17 line 6 (98.05.12)
ind$(a_1,a_2,\ldots,a_n)$ \becomes ind$(a_1\,a_2\ldots a_n)$
\endchange

\improvepage 3.18 line 1 (98.05.23)
\ninepoint $5\,4\,6\,1\,3\,8\,7\,2$; \becomes
$5\,4\,6\,1\,3\,8\,7\,2$ (man~1 is 5th out, etc.);
\endchange

\bugonpage 3.20 line 2 of exercise 19 (99.08.09)
\ninepoint $\bigl((n-1)m\mod m\bigr)$ \becomes $\bigl((n-1)m\mod n\bigr)$
\endchange

\amendpage 3.22 line 1 of exercise 28 (02.11.23)
\ninepoint (R. W. Floyd, 1983.) \becomes
\endchange

\improvepage 3.23 lines $-20$ and $-15$ (03.08.15)
{\sl Anuyogadv\=ara-sutra\/} \becomes {\sl Anuyogadv\=aras\=utra\/}
\endchange

\improvepage 3.23 near the bottom (00.07.20)
line $-12$: {\sl L\'\i\kern.05eml\'avat\'\i\/} of Bh\'ascara \'Ach\'arya
 \becomes {\sl L{\=\i}l\=avat{\=\i}\/} of Bh\=askara\nl
lines $-11$ and $-8$: Bh\'ascara \becomes Bh\=askara
\endchange

\amendpage 3.23 from line $-4$ and continuing into page 24 (04.10.15)
The correct rule \dots A few years later, \ \becomes\par
The correct rule for counting permutations when elements are repeated was
apparently unknown in Europe until Marin Mersenne stated it without proof
as Proposition 10 in his elaborate treatise on melodic principles
[{\sl Harmonie Universelle\/ \bf2}, also entitled {\sl Traitez de la Voix
et des Chants\/} (1636), 129--130].
Mersenne was interested in the number of tunes that could
be made from a given collection of notes; he observed, for example, that
a theme by Boesset,
$$\vcenter{\epsfxsize=2in\epsfbox{\figdir/mersenne-tune.eps}}$$
can be rearranged in exactly $15!/(4!@@3!@@3!@@2!)=756{,}756{,}000$ ways.

The general rule \eq(3) also appeared in Jean Prestet's {\sl \'El\'emens des
Math\'e\-ma\-tiques\/} (Paris: 1675), 351--352, one of the very first
expositions of combinatorial mathematics to be written in the Western world.
Prestet stated the rule correctly for a general multiset, but illustrated
it only in the simple case $\{a,a,b,b,c,c\}$.
A few years later,
\endchange

\improvepage 3.27 line $-10$ (02.08.22)
{\sl Paris\/ \bf258} (1964) \becomes {\bf258} (Paris, 1964)
\endchange

\improvepage 3.31 line 2 of exercise 10 (03.06.15)
\ninepoint insure \becomes ensure
\endchange

\bugonpage 3.31 line $-5$ (01.07.20)
\ninepoint notion of \becomes notion of ``topological sorting''
\endchange

\bugonpage 3.32 line 3 of exercise 15 (02.08.13)
\ninepoint $x_1<x_2<\cdots <x_n$ and $n_1+n_2+\cdots+n_m=m$ \becomes
           $x_1<x_2<\cdots <x_m$ and $n_1+n_2+\cdots+n_m=n$
\endchange

\improvepage 3.33 line 8 of exercise 19 (06.01.01)
\ninepoint
$\pi=x_{j_1}\,x_{j_2}\ldots x_{j_n}$, where $1\le j_k\le m$ \becomes
$\pi=x_{i_1}\,x_{i_2}\ldots x_{i_n}$, where $1\le i_k\le m$
\endchange

\bugonpage 3.35 line 2 (06.01.01)
\ninepoint the largest $x_j$ \becomes the largest $j$
\endchange

\improvepage 3.35 line 7 (06.01.01)
\ninepoint $\circR$ is simply juxtaposition; \becomes
$\circR$ sorts $\{w_1,\ldots,w_k,x_1,\ldots,x_l\}$, but it simply
juxtaposes $y_1\ldots y_k$ with $z_1\ldots z_l$;
\endchange

\improvepage 3.35 replacement for the displayed equation on line 10 (06.01.01)
\ninepoint
$$\pi=(x_{11}\ldots x_{1n_1}\,y_1)\circR\bigl((x_{21}\ldots x_{2n_2}\,y_2)
\circR\bigl({}\cdots\circR(x_{t1}\ldots x_{tn_t}\,y_t)\cdots{}\bigr)\bigr)$$
\endchange

\improvepage 3.35 line 11 (06.01.01)
\ninepoint by letting \eq(11) stand for \becomes
by letting the two-line array \eq(11) correspond to
\endchange

\bugonpage 3.39 line 5 (06.06.13)
{\sl Magazine\/ \bf33} \becomes
{\sl Magazine\/ \bf32}
\endchange

\bugonpage 3.40 lines 10 and 11 from the bottom (00.05.18)
$\displaystyle{}+\sum_{j=0}^k$ \becomes
$\displaystyle{}+\sum_{j=0}^{k-1}$ \qquad (twice)
\endchange

\bugonpage 3.41 replacement for the bottom line (00.05.19)
$$\def\z{\mskip2mu\overline{\mskip-2mu z\?}@}
\medmuskip2mu
R_m(z)=L(z)+{2z\over z-z_0}+{z\over z-z_1}
+{z\over z-\z _1}+{z\over z-z_2}+ {z\over z-\z _2}
+\cdots +{z\over z-z_m}+{z\over z-\z _m}$$
\endchange

\bugonpage 3.42 replacement for line 5 (00.05.19)
$$\def\z{\mskip2mu\overline{\mskip-2mu z\?}@}
{2z\over 1-z}+{z/z_1\over 1-z/z_1}+{z/\z_1\over 1-z/\z_1}+\cdots
+{z/z_m\over 1-z/z_m}+{z/\z_m\over 1-z/\z_m}+R_m(z),$$
\endchange

\bugonpage 3.42 line 3 after {\eq(31)} (00.05.19)
$R_m(z)\to-z$ \becomes $R_m(z)\to cz$ for some constant~$c$
\endchange

\amendpage 3.42 line 4 after {\eq(31)} (00.05.26)
when $n>1$. \becomes when $n>1$. (See also exercise 28.)
\endchange

\bugonpage 3.43 in Figure 3 (06.01.01)
[the $x$ axis, namely the horizontal line $y=0$, should be thicker]
\endchange

\improvepage 3.45 replacement for line 3 (00.05.18)
\ninepoint
$$\sum_k\,\Bigl\langle{n\atop k}\Bigr\rangle\,\Bigl({k\atop n-q}\Bigr)=
\Bigl\{{n\atop q}\Bigr\}\,q!, \qquad\hbox{integer $q\ge0$}.$$
\endchange

\improvepage 3.46 line 3 (05.01.29)
\ninepoint alternatively \becomes alternately
\endchange

\amendpage 3.46 exercise 16 (02.08.14)
\ninepoint[change the notation from $n\parts k$ to
$\rlap{$\bigr\rangle$}{n\euler k}\llap{$\bigl\langle$}$, because I'm
now using the former notation for partition numbers in Volume~4]
\endchange

\amendpage 3.46 exercise 19 (03.11.26)
\ninepoint J. Riordan \becomes I. Kaplansky and J. Riordan, 1946
\endchange

\bugonpage 3.47 line 1 of exercise 27 (06.01.01)
\ninepoint is a forest \becomes is an oriented forest
\endchange

\amendpage 3.47 new exercise (00.05.26)
\ninepoint
\ex28. [\HM35] Find the asymptotic value of the numbers $z_m$ in Fig.~3
as $m\to\infty$, and prove that
$$
\sum_{m=1}^\infty\bigl(z_m\_ + \bar z_m\_\bigr)\;=\;e-{5\over2}.
$$
\endchange

\amendpage 3.47 another new exercise (05.01.29)
\ninepoint
\EX29. [M30] The permutation $a_1\ldots a_n$ has a ``peak'' at $a_j$ if
$1<j<n$ and $a_{j-1}<a_j>a_{j+1}$. Let $s_{nk}$ be the number of permutations
with exactly $k$ peaks, and let $t_{nk}$ be the number with $k$ peaks {\it and\/}
$k$~descents. Prove that (a)~$s_{nk}=
  \half\rlap{$\bigr\rangle$}{n\euler2k}\llap{$\bigl\langle$}
+      \rlap{$\bigr\rangle$}{n\euler2k+1}\llap{$\bigl\langle$}
+ \half\rlap{$\bigr\rangle$}{n\euler2k+2}\llap{$\bigl\langle$}$
(see exercise~16);
(b)~$s_{nk}=2^{n-1-2k}t_{nk}$;
(c)~$\sum_k{n\euler k}x^k=\sum_k t_{nk}x^k(1+x)^{n-1-2k}$.
\endchange

\bugonpage 3.52 line 16 (98.05.12)
poof \becomes proof
\endchange

\improvepage 3.54 line 4 (06.01.01)
array \eq(16); interchanging \becomes
array \eq(16); its columns are
essentially independent and can be rearranged. Interchanging
\endchange

\bugonpage 3.59 line $-6$ (06.01.01)
in much \becomes in a much
\endchange

\bugonpage 3.60 line 4 after the caption (98.11.25)
{\medmuskip=.5\medmuskip
$(n_2+m-1)$ \becomes $(n_2+m-2)$}
\endchange

\improvepage 3.61 line $-6$ (06.01.01)
$a_j>a_k>a_i$ for $i<j<k$ \becomes
$a_i>a_k>a_j$ for $i<j<k$
\endchange

\bugonpage 3.63 line 1 of {\eq(46)} (99.05.17)
$\half \ln n$ \becomes $\half n\ln n$
\endchange

\bugonpage 3.63 line 2 of {\eq(49)} (99.08.09)
$\displaystyle{x^6\over n^2}$ \becomes $\displaystyle{8\over 9}{x^6\over n^2}$
\endchange

\amendpage 3.64 line 2 above {\eq(52)} (06.01.01)
$O\bigl(\exp(-2n^\epsilon)\bigr)$ in each term \becomes
$O\bigl(\exp(-2n^{2\epsilon})\bigr)$ in each term
\endchange

\improvepage 3.64 line $-6$ (02.08.22)
{\sl Paris\/} (1782) \becomes (Paris, 1782)
\endchange

\improvepage 3.67 replacement for line $-4$ (06.01.01)
\ninepoint
$$17!/(12\cdot 11\cdot 8\cdot 7\cdot 5\cdot 4\cdot 1\cdot 9\cdot 6\cdot 5\cdot
3\cdot 2\cdot 5\cdot 4\cdot 2\cdot 1\cdot 1)$$
[and similarly, insert parentheses into the denominator in exercise 20]
\endchange

\amendpage 3.68 exercise 23 (07.08.20)
\ninepoint $A_n$ \becomes $E_n$ \qquad(twice)
\endchange

\improvepage 3.69 exercise 32 (05.02.09)
\ninepoint $t_n$ is \becomes the involution number $t_n$ is
\endchange

\improvepage 3.70 line 6 (03.01.03)
\ninepoint outdegree \becomes out-degree
\endchange

\bugonpage 3.70 line 12 (06.01.01)
\ninepoint columns sets \becomes column sets
\endchange

\improvepage 3.74 hyphenation on lines $-8$ and $-7$ (07.10.09)
rear-ranged \becomes re-arranged
\endchange

\bugonpage 3.76 line $-15$ (98.07.21)
of other \becomes of the other
\endchange

\bugonpage 3.77 top line (98.06.07)
{\tenssbx Table 2} \becomes {\tenssbx Table 1}
\endchange

\improvepage 3.77 line $-4$ (98.06.03)
$K_1$, \dots, $K_{\!N}$ \becomes $K_1\,\ldots\,K_{\!N}$
\endchange

\bugonpage 3.82 replacement for line 11 (98.05.01)
$$B=\bigl(\min 0,\;\hbox{ave}\;(N^2-N)/4,\;{\max}\;(N^2-N)/2,\;\hbox{dev}\;
\sqrt{N(N-1)(N+2.5)}/6\bigr);$$
\endchange

\bugonpage 3.84 line 4 after the table (98.05.23)
that w sort \becomes that we sort
\endchange

\bugonpage 3.90 line 11 (98.08.09)
$h_t,\ldots,h_1$ \becomes $h_{t-1},\ldots,h_0$
\endchange

\bugonpage 3.92 in {\eq(9)} (00.05.22)
$\displaystyle{r-1\choose2}\le s\le{r\choose2}$ \becomes
$\displaystyle\Bigl({r-1\atop2}\Bigr)\le s<\Bigl({r\atop2}\Bigr)$
\endchange

\bugonpage 3.92 line 1 of Theorem I (98.12.09)
$O(Ne^{@c\sqrt{@\ln n}})$ \becomes $O(Ne^{@c\sqrt{@\ln N}})$
\endchange

\amendpage 3.94 new entry before line $-3$ of Table 6 (01.04.11)
\rightline{\ninerm
190 \ 84 \ 37 \ 16 \ 7 \ 3 1 \ \ 359 \ \ \ 7201 \ \ 7\qquad}
\endchange

\amendpage 3.94 near the bottom (01.04.11)
line $-5$: $10NT$ \becomes $10(NT-S)$\nlh
bottom three lines: Therefore \dots~first pass.\becomes
(The first pass is very
quick, however, if $h_{t-1}$ is near~$N$, because
$NT-S=(N-h_{t-1})+\cdots+(N-h_0)$.)
\endchange

\amendpage 3.95 lines 6--9 (01.04.11)
He also discovered \dots~order $N^{3/4}$. \becomes
On the other hand, subsequent tests by Marcin Ciura show that Sedgewick's
sequence~\eq(11) apparently makes
$B_{\rm ave}=O\bigl(N(\log N)^2\bigr)$ or better. The standard
deviation for sequence \eq(11) is amazingly small for $N\le10^6$, but it
mysteriously begins to ``explode'' when $N$ passes~$10^7$.
\endchange

\amendpage 3.95 near the bottom (01.04.11)
in \eq(12): $3h_t\ge N$ \becomes $h_{t+1}>N$\nlh
lines $-6$ to $-4$: recommended, again \dots~10\% faster. \becomes
recommended. Still better results,
possibly even of order $N\log N$, have been reported by N.~Tokuda
using the quantity $\lfloor2.25h_s\rfloor$ in place of
$3h_s$ in~\eq(12); see {\sl Information Processing 92\/ \bf1} (1992),
449--457.
\endchange

\improvepage 3.98 line $-1$ (00.12.26)
discussed below in \becomes discussed in
\endchange

\bugonpage 3.102 line 2 of exercise 7 (99.05.12)
\ninepoint $\left\vert a_2-1\right\vert$ \becomes $\left\vert a_2-2\right\vert$
\endchange

\bugonpage 3.103 line 2 of exercise 17 (05.01.24)
\ninepoint $\{1,2,\ldots,n\}$ \becomes $\{1,2,\ldots,N\}$
\endchange

\bugonpage 3.105 line 4 (00.05.22)
\ninepoint the running time \becomes the average running time
\endchange

\improvepage 3.105 line 2 of exercise 40 (02.11.01)
\ninepoint (15) \becomes \eq(15)
\endchange

\bugonpage 3.105 line 4 of exercise 42 (00.05.22)
\ninepoint $N/g$ \becomes $N^{3/2}\!/g$
\endchange

\amendpage 3.110 caption to Figure 16 is not an error (02.08.22)
\ninepoint cocktail-shaker short [shic] \becomes cocktail-shaker short [shic]
\vskip1pt\noindent\eightpoint
[Let's drink a toast to all alcoholic shorting methods, and forgive authors
who occasionally make weird attempts at humor.]
\endchange

\amendpage 3.111 step M3 of Algorithm M (05.05.19)
`$@\wedge@$' \becomes `$@\band@$'\qquad(six changes)
\endchange

\bugonpage 3.116 line 2 of step Q7 (98.05.01)
$r+1$. If \becomes $r+1$.) If
\endchange

\improvepage 3.118 line 53 of the program (98.07.27)
\ninepoint
$j-N\!@$ \becomes $j-N\!@$.
\endchange

\improvepage 3.119 lines 21 and 22 (07.04.24)
of the keys $K_1,\ldots,K_s$ \becomes
of the first~$s$ keys $\{K_1,\ldots,K_s\}$
\endchange

\improvepage 3.121 replacement for line 2 of {\eq(25)} (98.08.15)
\centerline{$B_N={1\over 6}\,(N+1)\bigl(2H_{N+1}-2H_{M+2}+1-6/(M+2)\bigr)
  +\half,$}
\endchange

\bugonpage 3.122 line 7 (00.05.22)
Exercise 58 \becomes Exercise 42
\endchange

\bugonpage 3.125 line 1 of step R2 (98.08.15)
$R_l\le \cdots \le R_r$ \becomes $R_l\ldots R_r$
\endchange

\bugonpage 3.125 line $-3$ (99.08.15)
\ninepoint [$\rI4=l-j$] \becomes [$\rI4=j-l@$]
\endchange

\bugonpage 3.126 lines {\it 30\/}  and {\it 31\/} of the program (98.08.15)
\ninepoint [rI4unknown] \becomes [rI4 unknown]\nl
$b\gets b-1$ \becomes $b\gets b+1$
\endchange

\bugonpage 3.127 line $-15$ (00.09.14)
${1\over 4}(\alpha +1)N$ \becomes $\half N$
\endchange

\bugonpage 3.130 line 1 (01.09.21)
$f_{-\?1}$ \becomes $f_{-\?1}(x)$
\endchange

\amendpage 3.136 new sentence added to exercise 28 (98.10.25)
\ninepoint Ignore the comparisons made when computing the median value~$s$.
\endchange

\bugonpage 3.137 line 2 of exercise 41 (00.10.28)
\ninepoint $1\le k<i$ \becomes $l\le k<i$
\endchange

\bugonpage 3.138 line 2 of exercise 55 (99.08.05)
\ninepoint three keys \eq(28). \becomes
three keys \eq(28), assuming that $M>1$.
\endchange

\bugonpage 3.141 line $-18$ (98.05.23)
$\{$170,~175$\}$ \becomes $\{$170,~275$\}$
\endchange

\bugonpage 3.148 line 4 (00.09.14)
$\ln N$ \becomes $\lg N$
\endchange

\bugonpage 3.151 line $-7$ (98.05.01)
heaps). \becomes heaps.)
\endchange

\bugonpage 3.152 line 9 (00.01.09)
{\sl CACM\/ \bf12} \becomes {\sl CACM\/ \bf21}
\endchange

\amendpage 3.152 line 14 (00.07.29)
249 \becomes 249; M.~L. Fredman, {\sl JACM\/ \bf46} (1999), 473--501
\endchange

\amendpage 3.152 lines 24--25 (00.03.20)
 {\sl SODA\/ \bf8} (1997), 83--92] \becomes
 {\sl SICOMP\/ \bf28} (1999), 1326--1346]
\endchange

\bugonpage 3.155 line 2 (00.09.14)
04805$-$ \becomes 04806$-$
\endchange

\amendpage 3.163 line $-20$ (02.01.01)
regardless \becomes or in the trivial case $N=1$, regardless
\endchange

\bugonpage 3.166 line 4 after the program (05.12.04)
about $9N\lg N$ \becomes about $(8N\lg N)u$
\endchange

\bugonpage 3.166 line $-12$ (98.07.29)
111.] \becomes 111].
\endchange

\bugonpage 3.166 line $-8$ (98.08.13)
\ninepoint arrangement \becomes arrangements
\endchange

\bugonpage 3.166 line $-3$ (98.07.31)
\ninepoint $K_1'<\cdots K_N'$ \becomes $K_1'<\cdots<K_N'$
\endchange

\improvepage 3.166 line $-1$ (99.05.11)
\ninepoint key appears \becomes keys appear
\endchange

\improvepage 3.168 line 1 of exercise 18 (00.02.16)
\ninepoint on $N$ records \becomes of $N$ records
\endchange

\improvepage 3.174 line 07 of the program (99.10.05)
$\to\.{TOP[$i$]}$ \becomes $\to\.{TOP[$i$]}$.
\endchange

\bugonpage 3.177 line 1 (99.11.20)
sort if \becomes sort it
\endchange

\improvepage 3.177 line $-1$ (98.08.10)
\ninepoint {\it signed-magnitude\/} \becomes {\it signed magnitude}
\endchange

\bugonpage 3.183 line 23 (98.08.09)
$\lceil\ln n!\rceil$ \becomes $\lceil\lg n!\rceil$
\endchange

\bugonpage 3.188 row $b$ and column $e$ of {\eq(24)} (98.06.16)
9 \becomes 7
\endchange

\bugonpage 3.189 in Eq.~{\eq(26)} (00.10.28)
$\displaystyle{n'@!\over 2^{k'}T(G')}@ {n''@!\over 2^{k''}T(G')}$ \ \becomes\
$\displaystyle{n'@!\over 2^{k'}T(G')}@ {n''@!\over 2^{k''}T(G'')}$
\endchange

\amendpage 3.192 lines 3--5 (07.01.02)
The intermediate \dots~takes no fewer comparisions \becomes
Marcin Peczarski [see {\sl Algorithmica\/ \bf40} (2004), 133--145;
{\sl Information Proc.\ Letters\/ \bf101} (2007), 126--128]
extended Wells's method and proved that $S(13)=34$, $S(14)=38$, $S(15)=42$,
$S(22)=71$; thus merge
insertion is optimum in those cases as well. Intuitively, it seems likely that
$S(16)$ will some day be shown to be less than $F(16)$, since $F(16)$ involves
no fewer steps
\endchange

\amendpage 3.193 lines 2 and 3 after the figure (06.04.14)
(This result \dots~1956.) \becomes
[This result was first obtained by A.\ Gleason in an internal
IBM memorandum (1956).]
\endchange

\amendpage 3.194 line 1 (07.09.22)
\ninepoint When \becomes ({\it Weak orderings.}) When
\endchange

\amendpage 3.197 exercise 35 (02.08.20)
\ninepoint $S(13)$ \becomes $S(14)$
\endchange

\bugonpage 3.204 line 2 (99.10.05)
$M(m,n{-}2^t)$ \becomes $H(m,n{-}2^t)$
\endchange

\bugonpage 3.212 bottom line (99.01.04)
acieves \becomes achieves
\endchange

\bugonpage 3.213 line $-4$ (02.12.23)
exercise 26 \becomes exercise 27
\endchange

\bugonpage page 3.224 line 4 before {\eq(3)} (01.09.21)
$\langle y_1,\ldots,y_m\rangle$ \becomes $\langle y_1,\ldots,y_n\rangle$
\endchange

\bugonpage 3.225 line $-13$ (98.06.02)
needed in an \becomes needed in a
\endchange

\improvepage 3.225 line $-8$ (00.01.21)
still unknown. \becomes known only in a very weak sense.
\endchange

\bugonpage 3.226 line $-7$ (99.06.30)
Hughes \becomes Hugues
\endchange

\improvepage 3.227 replacement for bottom part of Fig.\ 49 (98.08.10)
$$\epsfbox{\figdir/ch5.494}$$
\endchange

\amendpage 3.229 new sorting networks for Fig.\ 51 (01.10.11)
\smallskip\noindent
\line{\valign{\vfil#\vfil\cr
\epsfbox{\figdir/ch5.511}\cr
\noalign{\hfil}
\epsfbox{\figdir/ch5.515}\cr}}
\endchange

\amendpage 3.229 line $-10$ (01.10.11)
$n=6$ and $n=9$ \becomes $n=6$, $n=9$, and $n=11$
\endchange

\amendpage 3.229 replacement for the bottom two lines (01.10.11)
they always sort. Some of these networks were discovered in 1969--1971
by G.~Shapiro ($n=6$, 12) and D.\ Van Voorhis ($n=10$, 16);
the others were found in 2001 by
Loren Schwiebert, using genetic methods ($n=9$, 11).
\endchange

\bugonpage 3.230 line 22 (01.09.21)
$\{z_{2n+1},\ldots,z_{4n}\rangle$ to $\{z_1,\ldots,z_{2n}\rangle$ \becomes
$\{z_{2n+1},\ldots,z_{4n}\}$ to $\{z_1,\ldots,z_{2n}\}$
\endchange

\improvepage 3.231 line $-5$ (02.12.23)
$(m,n)$ merging \becomes $(m,n)$-merging
\endchange

\improvepage 3.232 line $-22$ (03.01.24)
$(1,n)$ merging \becomes $(1,n)$-merging
\endchange

\amendpage 3.235 notation change, four lines before exercise 1 (05.08.08)
\ninepoint $x\le y$ \becomes $x\subseteq y$\qquad (twice)
\endchange

\amendpage 3.238 replacement for exercise 20 (03.09.12)
\ninepoint
\ex 20. [28] Prove that (a) $\hat V_3(5)=7$; (b) $\hat U_4(n)\le3n-10$ for
$n\ge6$.
\endchange

\amendpage 3.238 notation change in line 2 of exercise 22 (05.08.08)
\ninepoint
$x\le y$ implies that $x\alpha\le y\alpha$ \becomes
$x\subseteq y$ implies that $x\alpha\subseteq y\alpha$
\endchange

\bugonpage 3.240 line $-5$ (98.08.27)
\ninepoint $r\le4n^2+\sqrt n\,\lg n$ \becomes $r\le4n^2+O(n^{3/2}\log n)$
\endchange

\improvepage 3.241 line 1 of exercise 43 (02.12.23)
\ninepoint $(m,n)$ merging \becomes $(m,n)$-merging
\endchange

\bugonpage 3.241 two lines above Fig.\ 59 (04.08.30)
replace comparisons by $m$-way merges. \becomes
replace each comparison by a merge network with $\hat M(m,m)$ modules.
\endchange

\improvepage 3.242 lines $-3$, $-13$, $-17$ (98.08.17)
\ninepoint
Fig.\ 61 \becomes Fig.\ 60\nl
{\bf Fig.\ 61} \becomes {\bf Fig.\ 60}
\endchange

\bugonpage 3.246 line 2 of exercise 66 (99.10.05)
\ninepoint exercise 63 \becomes exercise 64
\endchange

\improvepage 3.246 lines $-1$, $-3$ (98.08.17)
\ninepoint
{\bf Fig.\ 60} \becomes {\bf Fig.\ 61}\nl
Fig.\ 60 \becomes Fig.\ 61
\endchange

\bugonpage 3.247 line 4 (99.10.29)
\ninepoint $1\le i\le N$ \becomes $1<i\le N$
\endchange

\bugonpage 3.250 line 22 (03.11.21)
15,000000 \becomes 15000000
\endchange

\bugonpage 3.262 line $-13$ (03.01.10)
685--687 \becomes 685--688
\endchange

\improvepage 3.264 line 2 of exercise 23 (99.05.11)
\ninepoint the same order that \becomes the same order as
\endchange

\improvepage 3.268 line 10 (05.02.14)
\ninepoint the `2,2,2' on T3 in Phase 3 should be right-justified
\endchange

\amendpage 3.270 lines 5 and 6 (04.09.21)
the usual Fibonacci sequence; \dots~by V.~Schlegel \becomes
the usual Fibonacci sequence,
and when $p=3$ it has been called the Tribonacci sequence.
Such sequences were apparently first studied for $p>2$ by
N\=ar\=aya\d{n}a Pa\d{n}\d{d}ita in~1356 [see P.~Singh, {\sl Historia
Mathematica\/ \bf12} (1985), 229--244], then many years later by
\endchange

\bugonpage 3.271 line 1 of step D1 (98.05.06)
$\.{D[$@j$]}-1$ and $\.{TAPE[$k$]}\gets j$ \becomes
$\.{D[$@j$]}\gets1$ and $\.{TAPE[$@j@$]}\gets j$
\endchange

\improvepage 3.275 equation number in wrong font (99.11.14)
\ninepoint\eq(12) \becomes \tenpoint\eq(12)
\endchange

\bugonpage 3.282 line $-10$ (99.11.14)
$v_1+u_0+v_0$ \becomes $v_1$
\endchange

\improvepage 3.286 line 3 of exercise 14 (99.11.14)
\ninepoint $T_{(n(k)+1),k}$ \becomes $T_{(n(k)+1)k}$
\endchange

\bugonpage 3.287 the running headline (02.07.21)
{\eightrm CASCADE \becomes POLYPHASE}
\endchange

\amendpage 3.288 line 3 (03.06.19)
{\sl National Conf.} \becomes {\sl Nat.\ Meeting}
\endchange

\bugonpage 3.294 line 7 of {\eq(4)} (01.02.08)
$d_n=c_n-e_{n-1}$ \becomes $d_n=c_n-c_{n-1}$
\endchange

\improvepage 3.296 line $-6$ (03.04.08)
Eqs.~\eq(8) \becomes The equations in \eq(8)
\endchange

\bugonpage 3.299 line $-18$ (00.01.16)
Programmer \becomes Programmers
\endchange

\amendpage 3.316 lines 1 and 2 (05.12.04)
unless successfully contested, such a patent makes it illegal to use \becomes
between 1968 and 1988, no one in the U.S.A. could legally use
\endchange

\bugonpage 3.316 line 2 of exercise 2 (05.02.14)
\ninepoint 11 dummy runs \becomes 10 dummy runs
\endchange

\bugonpage 3.332 line 17 (00.10.28)
$nC(1+\rho)@@\omega_o\tau$ \becomes $NC(1+\rho)@@\omega_o\tau$
\endchange

\improvepage 3.345 lines $-20$, $-19$, and $-18$ (04.09.17)
reflection shows why \dots~into $S$ runs. \becomes
reflection shows why this is so, if we consider the actions of
a merge sort and imagine that time could run backwards: The final output
is ``unmerged'' into subfiles, which are
unmerged into others, etc.; at time zero the output has been unmerged into
$S$ runs.
\endchange

\improvepage 3.348 line 3 of exercise 1 (98.08.10)
\ninepoint mixed radix system \becomes mixed-radix number system
\endchange

\bugonpage 3.348 line $-13$ (98.05.18)
preform \becomes perform
\endchange

\improvepage 3.350 in the specification of \.{SORT10} (03.01.10)
Tape 0 \becomes tape 0
\endchange

\bugonpage 3.356 line 1 of exercise 6 (03.01.10)
\ninepoint Fig.~87 \becomes Fig.~88
\endchange

\improvepage 3.360 line $-6$ (04.09.17)
operations would take \becomes operations might take
\endchange

\amendpage 3.369 replacement for the final three paragraphs (03.07.17)
SyncSort begins by reading the first block of each run and putting these
$P\?B$ records into the memory pool. Each record in the memory pool is linked
to its successor in the run it belongs to, except that the final record in
each block has no successor as yet. The smallest of the keys in those
final records determines the run that will need to replenished first, so we
begin to read the second block of that run into the first buffer.
Merging begins as soon as that
second block has been read; by looking at its final key we can accurately
forecast the next relevant block, and we can continue in the same way
to prefetch exactly the right blocks to input, just before they are needed.

The three SyncSort buffers are arranged in a circle. As merging proceeds, the
computer is processing data in the current buffer, while input is being read
into the next buffer and output is being written from the third. 
The merging algorithm exchanges each record in the current buffer
with the next record of output, namely the record in the memory pool that has
the smallest key. The selection tree and the successor links are also updated
appropriately as we make each exchange. Once the end of the current buffer is
reached, we are ready to rotate the buffer circle: The reading buffer becomes
current, the writing buffer is used for reading, and we begin to write from
the former current buffer.

Many extensions of this basic idea are possible, depending on hardware
capabilities. For example, we might use two disks, one for reading and one
for writing, so that input and output and merging can all take place
simultaneously. Or we might be able to overlap seek time by
extending the circle to four or more
buffers, as in Fig.~26 of Section~1.4.4, and deviating from the
forecast input order.
\endchange

\improvepage 3.370 line 11 (05.12.04)
$D$ reads or writes \becomes $D$ reads or $D$ writes
\endchange

\amendpage 3.370 line 16 (04.05.03)
faster. \becomes faster unless they're being done simultaneously.
\endchange

\bugonpage 3.371 line $-20$ (04.05.03)
$b_1$ \becomes $h_1$
\endchange

\bugonpage 3.377 line 6 of exercise 7 (00.10.28)
\ninepoint $6w_1+6w_2+7w_3+9w_4+9w_5+7w_3+4w_7+4w_8$ \becomes
$6w_1+6w_2+7w_3+9w_4+9w_5+7w_6+4w_7+4w_8$
\endchange

\improvepage 3.383 lines $-12$ and $-11$ (05.09.20)
at the beginning of each decade \becomes every ten years
\endchange

\amendpage 3.385 line 16 (02.02.18)
1930 \becomes 1929--1930
\endchange

\bugonpage 3.385 line 24 (07.08.22)
(1938) \becomes (1936)
\endchange

\bugonpage 3.385 line 27 (07.08.22)
James W. Bryce, {\sl U.S.\ Patent 2189024\/} (1940) \becomes
Ralph E. Page, {\sl U.S.\ Patent 2359670\/} (1944)
\endchange

\improvepage 3.385 line $-16$ (00.12.04)
stored program computer \becomes stored-program computer
\endchange

\amendpage 3.386 near the bottom (03.01.25)
line $-9$: Holberton \becomes Snyder\nl
line $-1$: Mrs.~Holberton \becomes Snyder
\endchange

\amendpage 3.387 line 22 (03.01.25)
F. E. Holberton \becomes Frances E. [Snyder] Holberton
\endchange

\amendpage 3.389 line $-15$ (99.07.03)
{\sl SODA\/ \bf8} (1997), 370--379 \becomes
{\sl J.~Algorithms\/ \bf31} (1999), 66--104.
\endchange

\bugonpage 3.400 equation {\eq(12)} (98.06.23)
$c_1=1/N^\theta$ \becomes $c=1/N^\theta$
\endchange

\bugonpage 3.400 equation {\eq(12)} (02.11.20)
$=0.1386$ \becomes $\approx0.1386$
\endchange

\bugonpage 3.403 line 17 (98.06.02)
a interesting \becomes an interesting
\endchange

\improvepage 3.406 line 14 (99.11.14)
\ninepoint $P_{N-1,m-1}$ \becomes $P_{(N-1)(m-1)}$
\endchange

\amendpage 3.407 line 1 of exercise 17 (05.05.23)
\ninepoint W. E. Smith \becomes J. R. Jackson
\endchange

\improvepage 3.410 line $-16$ (03.01.10)
$R_1@R_2\ldots R_N$ \becomes $R_1$, $R_2$, \dots, $R_N$
\endchange

\improvepage 3.412 equation number in wrong font (99.11.14)
\eightpoint\eq(1) \becomes \tenpoint\eq(1)
\endchange

\improvepage 3.414 line $-9$ (99.11.14)
number \ of \becomes number of
\endchange

\bugonpage 3.416 lines 5 and 6 after the program (99.11.14)
C1 is weighted more heavily than C2 \becomes
$C1$ is weighted more heavily than~$C2$
\endchange

\improvepage 3.418 line 7 (03.01.10)
$R_1@R_2\ldots R_N$ \becomes $R_1$, $R_2$, \dots, $R_N$
\endchange

\bugonpage 3.424 line 7 of exercise 20 (04.08.28)
\ninepoint Fibonacci search \becomes Fibonaccian search
\endchange

\improvepage 3.434 line $-13$ (04.05.03)
{\sl $\.S\gets \.{LLINK(R)}$} \becomes $\.S\gets \.{LLINK(R)}$
\endchange

\bugonpage 3.435 line 3 (00.01.19)
about $n^2$ \becomes about $N^2$
\endchange

\bugonpage 3.440 bottom line (04.06.28)
4.55 \becomes 4.72
\endchange

\improvepage 3.443 equation numbers in wrong font (99.12.12)
(20) \becomes \eq(20)\nl
(21) \becomes \eq(21)
\endchange

\bugonpage 3.446 line $-8$ (98.05.13)
in a several \becomes in several
\endchange

\improvepage 3.447 near the top (98.12.03)
line 4: cost $c$ \becomes weight $w$\nl
line 6: $-c$ \becomes $-w$\nl
line 7: it has been \becomes an optimum tree has been\nlh
lines 8 and 9: a sequence of such transformations will make $l_k\le l_{k+1}$
\becomes we have found an optimum tree in which $l_k=l_{k+1}$
\endchange

\bugonpage 3.447 at the end of the proof of Lemma X (98.12.03)
line $-4$ of the proof: $\ninepoint\rect(j{+}\?1)$ \becomes
                        $\ninepoint\rect(j{-}\?1)$ \qquad (twice)\smallskip\nl
line $-3$ of the proof: $\ninepoint\rect(i{+}\?1)$ \becomes
                        $\ninepoint\rect(i{-}\?1)$\smallskip\nl
line $-2$ of the proof: $\ninepoint\rect(k{+}\?1)$ \becomes
                        $\ninepoint\rect(k{-}\?1)$
\endchange

\bugonpage 3.447 replacementment for lines 22--28 (04.07.25)
\indent
Suppose $l_s<l_k-1\le l_{s+1}$ for some $s$ with $j\le s<k-1$. Let $t$~be the
smallest index $<k$ such that $l_t=l_k$. Then $l_i=l_k-1$ for $s<i<t$,
and ${}\!\ninepoint\rect(s{+}\?1)$ is a left~child; possibly $s+1=t$.
Furthermore $\ninepoint\sq(t)$ and $\ninepoint\rect(t{+}\?1)$ are siblings.
Replace their parent by $\ninepoint\rect(t{+}\?1)$;
replace $\ninepoint\sq(i)$ by $\ninepoint\rect(i{+}\?1)$ for $s<i<t@$; and
replace the~external node $\ninepoint\sq(s)$ by an internal node whose children are $\ninepoint\sq(s)$ and $\ninepoint\rect(s{+}\?1)$.
This change increases the cost by $\le q_s-q_t-q_{t+1}\le q_s-q_{k-1}-q_k$,
so it is an improvement if $q_s<q_{k-1}+q_k$. Therefore, by (iii), 
$l_j\ge l_k-1$.
\endchange

\bugonpage 3.447 line $-9$ (04.07.25)
(b) \becomes (a) or (b)
\endchange

\bugonpage 3.447 replacement for line $-5$ (04.07.25)
$$C(q_0',\ldots,q_{n-1}')\;\le\;C(q\losub0,\ldots,q\losub n)-(q_{k-1}\?+\?q_k).
\eqno(29)$$
\endchange

\bugonpage 3.447 replacement for line $-1$ (04.07.25)
$$\ninepoint\sq(0)\quad\ldots\quad\rect(j{-}\?1)\quad\rect(k{-}\?1)\quad\sq(k)\quad\sq(j)\quad
\ldots\quad\rect(k{-}2)\quad\rect(k{+}\?1)\quad\ldots\quad\sq(n).\eqno(30)$$
\endchange

\bugonpage 3.449 near the top (98.12.03)
line 4: $\ninepoint\rect(k{-}2)$ \becomes $\ninepoint\rect(s{-}\?1)$\nl
line 6: $j<i<k-1$ \becomes $j<i<s$.
\endchange

\bugonpage 3.450 in Fig.\ 19 (04.09.15)
[On the path from the root to external node \.G, the level numbers
on internal nodes should be $(0,1,2,3,4,5)$, not $(0,1,2,2,3,4)$.]
\endchange

\bugonpage 3.451 replacement for steps G1, G2, and G3 (04.09.18)
\algindentset{G1}%
\algstep G1. [Begin phase 1.] Set $\.{WT(X$_k$)}\gets w_k$ and
$\.{LLINK(X$_k$)}\gets \.{RLINK(X$_k$)}\gets\Lambda$ for $0\le k\le n$.
Also set $\.P_0\gets\.X_{2n+1}$, $\.{WT(P$_0$)}\gets
\infty$, $\.P_1\gets\.X_{@0}$, $t\gets1$, $m\gets n$.
Then perform step~G2 for $r=1$, 2, \dots,~$n$, and go to~G3.

\algstep G2. [Absorb $w_r$.] (At this point we have the basic condition
$$\.{WT(P$_{i-1}$)}\;>\;\.{WT(P$_{i+1}$)}\qquad\hbox{for $1\le i<t$};
\eqno(31)$$
in other words, the weights in the working array are ``$@$2-descending.'')
If $\.{WT(P$_{t-1}$)}\le w_r$, set $k\gets t$, perform Subroutine~C below,
and repeat step~G2.
Otherwise set $t\gets t+1$ and $\.P_t\gets\.X_r$.

\algstep G3. [Finish phase 1.] While $t>1$, set $k\gets t$ and perform
Subroutine~C below.
\endchange

\bugonpage 3.452 replacement for Subroutine C (04.09.18)
\algbegin Subroutine C (Combination). This recursive subroutine is the
heart of the Garsia\with Wachs algorithm.
It combines two weights, shifts them left as
appropriate, and maintains the 2-descending condition~\eq(31).
Variable~$j$ is local, but variables $k$, $m$, and $t$ are global.

\algstep C1. [Create a new node.] (At this point we have $k\ge2$.) Set $m\gets
m+1$, $\.{LLINK(X$_m$)}\gets\.P_{k-1}$, $\.{RLINK(X$_m$)}\gets\.P_k$,
$\.{WT(X$_m$)}\gets\.{WT(P$_{k-1}$)}+\.{WT(P$_k$)}$.

\algstep C2. [Shift the following nodes left.] Set $t\gets t-1$, then
$\.P_j\gets\.P_{j+1}$ for $k\le j\le t$.

\algstep C3. [Shift the preceding nodes right.] Set $j\gets k-2$; then while
$\.{WT(P$_j$)}<\.{WT(X$_m$)}$ set $\.P_{j+1}\gets\.P_j$ and $j\gets j-1$.

\algstep C4. [Insert the new node.] Set $\.P_{j+1}\gets\.X_m$.

\algstep C5. [Done?] If $j=0$ or $\.{WT(P$_{j-1}$)}>\.{WT(X$_m$)}$,
exit the subroutine.

\algstep C6. [Restore \eq(31).] Set $k\gets j$, $j\gets t-j$, and
call Subroutine~C recursively. Then reset $j\gets t-j$ and return to
step~C5.\quad\slug
\endchange

\bugonpage 3.453 line 22 (99.05.11)
It it is smaller, \becomes If it is smaller,
\endchange

\amendpage 3.454 line 7 before the exercises (05.10.05)
See also the paper \becomes
Further properties have been found by M.~Karpinski,
L.~L. Larmore, and W.~Rytter, {\sl Theoretical Comp.\ Sci.\ \bf180}
(1997), 309--324.
See also the paper
\endchange

\bugonpage 3.455 line 2 of exercise 11 (99.05.11)
\ninepoint empirical data \becomes
\endchange

\improvepage 3.457 line 1 of exercise 28 (99.05.11)
\ninepoint a ``optimum binary search'' \becomes an ``optimum binary search''
\endchange

\bugonpage 3.459 line 3 (99.07.28)
{\sl Doklady Akademi\t{\i}a Nauk\/} \becomes {\sl Doklady Akademii Nauk}
\endchange

\bugonpage 3.460 line 2 of Theorem A (98.12.08)
1.4404 \becomes 1.4405
\endchange

\improvepage 3.465 line 12 (03.02.16)
$\.K\!@$. \becomes $K\!@$.
\endchange

\improvepage 3.465 line $-2$ (99.11.28)
\ninepoint $-a$, 0 or 0 \becomes $-a$, 0, or 0
\endchange

\improvepage 3.466 line 17 (99.11.28)
\ninepoint $-a$, 0 or 0 \becomes $-a$, 0, or 0
\endchange

\bugonpage 3.469 lines $-3$ and $-2$ (98.06.02)
$.143+.153+.143+.143=.582$ \becomes $.143+.152+.143+.143=.581$
\endchange

\improvepage 3.470 line $-6$ (00.12.27)
3.3 \becomes 3
\endchange

\bugonpage 3.471 line $-20$ (00.12.27)
Section 5.2.2 \becomes Section 6.2.2
\endchange

\bugonpage 3.473 line 5 (00.10.28)
Set $\.P=\.R$ \becomes Set $\.P\gets\.R$
\endchange

\bugonpage 3.475 lines 3 and 4 after the caption (98.10.23)
both (although \becomes both, although
\endchange

\improvepage 3.476 line 9 (06.03.22)
many rebalancings \becomes several rebalancings
\endchange

\bugonpage 3.477 line $-5$ (99.11.28)
(1989) \becomes (1990)
\endchange

\amendpage 3.478 bottom line (98.08.19)
{\sl Lecture Notes in Comp.\ Sci.\ \bf1136} (1996), 91--106 \becomes
{\sl JACM\/ \bf45} (1998), 288--323
\endchange

\amendpage 3.479 line 1 of exercise 7 (07.06.04)
\ninepoint (N.\ J.\ A.\ Sloane and A.\ V.\ Aho.) \becomes
(A. V. Aho and N.\ J.\ A.\ Sloane.)
\endchange

\bugonpage 3.487 line 19 (00.12.08)
contain $m$ keys \becomes contains $m$ keys
\endchange

\bugonpage 3.489 line 27 (03.02.16)
121--138 \becomes 121--137
\endchange

\bugonpage 3.492 line 12 (03.02.16)
490--500 \becomes 490--499
\endchange

\bugonpage 3.493 at left of the table (99.11.28)
between \.I and \.J: \.{\char2} \becomes \.{\char1}\nl
between \.R and \.{\char5}: \.{\char8} \becomes \.{\char6}
\endchange

\amendpage 3.496 new paragraph to follow line 23 (01.04.07)
\indent
An interesting way to store large, growing tries in external memory was
suggested by S.~Y. Berkovich in {\sl Doklady Akademii Nauk SSSR\/ \bf 202}
(1972), 298--299 [English translation in {\sl Soviet Physics--Doklady
\bf 17} (1972), 20--21].
\endchange

\bugonpage 3.500 line 9 (98.09.11)
See $n$ \becomes Set $n$
\endchange

\improvepage 3.505 line $-5$ (99.11.28)
node.) \becomes node).
\endchange

\bugonpage 3.511 line 6 of exercise 41 (04.10.31)
\ninepoint that can written \becomes that can be written
\endchange

\amendpage 3.512 new exercise (99.03.08)
\ninepoint
\EX45. [M25] If the seven keys of Fig.~33 are inserted in random order by the
algorithm of exercise~15, what is the probability of obtaining the
tree shown?
\endchange

\bugonpage 3.513 line $-15$ (00.02.25)
{\sl Mecmuasi\/} \becomes {\sl Mecmuas\i\/}
\endchange

\bugonpage 3.514 lines 12 and 13, in column {\tt HER} (01.09.21)
$-1$ \becomes $1$
\endchange

\bugonpage 3.514 line 10 (99.10.15)
\.{K(3,3)} \becomes \.{K(3:3)}
\endchange

\bugonpage 3.515 line $-14$ (98.07.17)
good good spread \becomes good spread
\endchange

\bugonpage 3.517 in 13th and 14th printings only (03.11.14)
{\ninepoint[The following line was missing at the bottom of the page.]}\nl
\line{$h(2)$,~\dots, so
the following experiment suggests itself: Starting with the line}
\endchange

\amendpage 3.518 lines 6--8 (03.04.07)
was first conjectured \dots~proof.] \becomes
was observed long ago by botanists Louis and Auguste Bravais, {\sl Annales
des Sciences Naturelles\/ \bf7} (1837), 42--110, who gave an illustration
equivalent to Fig.~37 and related it to the Fibonacci sequence. See also
S.~\'Swierczkowski, {\sl Fundamenta Math.\ \bf 46} (1958), 187--189.]
\endchange

\improvepage 3.519 lines $-5$ and $-4$ (06.06.27)
Such arrays \dots~addition. \becomes
Such arrays make multiplication unnecessary.
If $M$ is a power of~2, we can avoid the
division in~\eq(9) by substituting exclusive-or for addition;
this gives a different, but equally good, hash function.
\endchange

\improvepage 3.523 line 2 after {\eq(13)} (99.12.19)
$\rA\equiv K$. \becomes $\rA\equiv K$; $\rI2\equiv\.{LINK[$i$]}$ and/or~$R$.
\endchange

\bugonpage 3.525 replacement for line 14 (01.09.21)
$$
\eqalignno{
C_N&=1+\phantom{N}{N-1\over 2M}\phantom{N}\approx 1+{\alpha\over2}
\quad\phantom{^2}\hbox{(successful search)}.&\eq(19)\cr}$$
\endchange

\improvepage 3.525 line 18 (04.08.02)
actually needed for \becomes that is occupied by
\endchange

\improvepage 3.525 line $-3$ (01.05.17)
open position \becomes empty position
\endchange

\improvepage 3.527 line 3 (03.01.03)
non-negative \becomes nonnegative
\endchange

\amendpage 3.529 line 3 of {\eq(24)} (05.05.19)
\ninepoint $\rA\gets\rA\vee1$ \becomes $\rA\gets\rA\bor1$
\endchange

\bugonpage 3.530 in {\eq(28)} (01.09.21)
{\medmuskip1mu
$M+1-n$ \becomes $M+1-N$}
\endchange

\bugonpage 3.532 bottom line (01.03.14)
$\.{TABLE[$(p_{@0}-c_1)\mod M$]}$ \becomes $\.{TABLE[$(p_1-c_1)\mod M$]}$
\endchange

\bugonpage 3.534 line 6 (98.12.17)
to step R2. \becomes to step R1.
\endchange

\amendpage 3.534 lines $-4$ and $-3$ (07.06.06)
exactly $r$ probes are needed \becomes any permutation of table locations
needs exactly $r$ probes
\endchange

\improvepage 3.537 line $-1$ (03.02.16)
Eq.~1.2.6--39 \becomes Eq.~1.2.6--\eq(39)
\endchange

\bugonpage 3.541 line 15 (01.09.21)
$\displaystyle\Bigl({M\atop N}\Bigr)$ \becomes
$\displaystyle\Bigl({M\atop n}\Bigr)$
\endchange

\amendpage 3.547 line $-9$ (03.02.13)
probing. \becomes probing. [See also Derr and Luke,
{\sl JACM\/ \bf3} (1956), 303.]
\endchange

\bugonpage 3.548 line $-5$ (98.07.07)
Witold Lipski \becomes Witold Litwin
\endchange

\bugonpage 3.550 line 3 (99.11.20)
\ninepoint
construct polynomial \becomes construct a polynomial
\endchange

\bugonpage 3.550 line 10 (01.09.21)
\ninepoint coefficients~$p_j$ \becomes coefficients~$p_i$
\endchange

\bugonpage 3.550 line $-4$ of exercise 8 (98.06.18)
\ninepoint
$\bigl\{(-1)^kq_k\theta\bigr\}$ \becomes $\bigl\{(-1)^{k+1}q_k\theta\bigr\}$
\endchange

\bugonpage 3.550 line $-3$ of exercise 8 (98.07.24)
\ninepoint
$\theta\bigr\}$ \becomes $\theta\bigr\}{}^+$
\endchange

\bugonpage 3.555 line 7 of exercise 55 (00.04.06)
\ninepoint Eq.~2.3.4.4--\eq(9) \becomes Eq.~2.3.4.4--\eq(21)
\endchange

\bugonpage 3.557 line 6 of exercise 72 (99.05.11)
\ninepoint than the expected \becomes then the expected
\endchange

\amendpage 3.558 new exercise (07.09.26)
\ninepoint
\EX78. [M26] (P. Woelfel.) If $0\le x<2^n$, let $h_{a,b}(x)=
\lfloor(ax+b)/2^k\rfloor\mod2^{n-k}$. Show that the set
$\{h_{a,b}\mid 0<a<2^n,\ \hbox{$a$ odd, and $0\le b<2^k$}\}$ is a
universal family of hash functions from $n$-bit keys to
$(n-k)$-bit keys. (These functions are particularly easy to implement
on a binary computer.)
\endchange

\bugonpage 3.561 line $-12$ (99.12.12)
for examples \becomes for example
\endchange

\bugonpage 3.565 line $-7$ (02.04.07)
$(k\mod l)+1$ \becomes $(l\mod k)+1$
\endchange

\improvepage 3.571 first two lines of\/ {\eq(11)} (99.11.20)
[semicolons are missing after the numeric values]
\endchange

\amendpage 3.571 line 15 (05.05.19)
$0010001000 \vee  0000100100 \vee  0000001001$ \becomes
$0010001000 \bor  0000100100 \bor  0000001001$
\endchange

\bugonpage 3.573 line $-15$ (99.11.17)
{\sl IEEE\./} \becomes {\sl IEEE\/}
\endchange

\bugonpage 3.574 line $-2$ (98.08.04)
1, 2, 5, 7, and 8 \becomes 0, 1, 4, 6, and 7
\endchange

\amendpage 3.575 line 18 (00.03.20)
In 1997, A. E. Brouwer found \becomes
A. E. Brouwer [{\sl SICOMP\/ \bf28} (1999), 1970--1971] has found
\endchange

\improvepage 3.575 line $-6$ (01.07.02)
stored three places \becomes stored thrice
\endchange

\bugonpage 3.577 line 3 (01.03.25)
triples, \becomes triples.
\endchange

\improvepage 3.578 line 22 (99.12.12)
{\sl Proc.$@$ACM\/} \becomes {\sl Proc.\ $\?$ACM\/}
\endchange

\bugonpage 3.578 line $-3$ (05.04.07)
{\sl J. Algorithms\/ \bf1} \becomes {\sl J. Algorithms\/ \bf2}
\endchange

\bugonpage 3.580 line 9 of exercise 8 (01.05.17)
\ninepoint
each of the queries \becomes each of the queries in
\endchange

\let\defaultpointsize=\ninepoint % get ready for answer pages

\bugonpage 3.584 line 6 (98.07.06)
{\eightss (1862) \becomes (1864)}
\endchange

\bugonpage 3.584 line 13 (02.08.24)
$p(1)\ldots p(n)$ and $q(1)\ldots q(n)$ \becomes
$p(1)\ldots p(N)$ and $q(1)\ldots q(N)$
\endchange

\bugonpage 3.584 line $-13$ (98.07.06)
$R_{p(n)}$ \becomes $R_{p(1)}$
\endchange

\bugonpage 3.587 last line of answer 12 (00.01.03)
$\lceil\lg n\rceil$ \becomes $\lceil\lg n\rceil+1$
\endchange

\improvepage 3.588 line 17 (98.09.14)
{\tt ACHTZEHNHUNDERT\]8ZWOLF\]8E} \becomes {\tt ACHTZEHNHUNDERTZWOLF\]8EIN}
\endchange

\improvepage 3.588 line 34 (98.11.11)
{\tt SUSSEN\]8MADE} \becomes {\tt LANGEN\]8TAG\]}
\endchange

\bugonpage 3.589 answer 19 (98.07.09)
line 1: $(x_i,x_j)$ \becomes $\{x_i,x_j\}$\nl
line 7: $(x_i,x_i)$ \becomes $\{x_i,x_i\}$\nl
lines 7 and 8: those of exercises 18 and 19 \becomes the method of exercise 18
\endchange

\amendpage 3.589 lines 10 and 11 of answer 19 (07.03.24)
Another approach \dots $c-x$). \becomes
Another approach, suggested by Jiang Ling, is to sort on a key such as
($x>c/2\Rightarrow x$, $x\le c/2\Rightarrow c-x$).
\endchange

\bugonpage 3.589 lines 2 and 3 of answer 20 (05.05.19)
`$@\wedge@$' \becomes `$@\band@$'\qquad(three changes)
\endchange

\bugonpage 3.589 line 2 of answer 21 (99.03.05)
\.{ERS}, \becomes \.{-ERS},
\endchange

\amendpage 3.589 bottom line (01.01.28)
Dudeney, \becomes
Dudeney, {\sl Strand\/ \bf65} (1923), 208, 312, and his
\endchange

\amendpage 3.590 line 11 (02.05.04)
$m$ is the computer word size. Sorting the file $\bigl(f(\alpha),\alpha\bigr)$
will \becomes
$m$ is, say, $2^{\lfloor 2\lg N\rfloor}$ when there are $N$ words.
Sorting the file $\bigl( f(\alpha),\alpha\bigr)$ with two passes of
Algorithm 5.2.5R will bring anagrams
\endchange

\improvepage 3.590 last line of answer 22 (02.08.24)
142.] \becomes 142].
\endchange

\bugonpage 3.592 line 6 of answer 7 (00.10.28)
Rodriguez \becomes Rodrigues
\endchange

\amendpage 3.592 lines 6 and 7 of answer 7 (01.12.19)
240; the $C$ inversion table appears in \becomes
240. The $C$ inversion table was used by Rothe in 1800; see also
\endchange

\bugonpage 3.593 answer 13 (07.06.02)
line 1: $b_{m-1}$, $b_{m+1}$ \becomes $b_{n-m}$, $b_{n-m+2}$\nl
line 2: $b_m$ \becomes $b_{n-m+1}$
\endchange

\amendpage 3.594 line 1 (02.12.11)
Euler's \becomes {\sl Comptes Rendus Acad.\ Sci.\
\bf92} (Paris, 1881), 448--450. Euler's
\endchange

\amendpage 3.594 line 1 of answer 20 (02.08.22)
See \becomes See J. J. Sylvester, {\sl Amer.\ J. Math.\ \bf5} (1882),
251--330, {\bf6} (1883), 334--336, \S57--\S68;
\endchange

\bugonpage 3.594 line 1 of answer 20 (00.01.07)
Zolnowski \becomes Zolnowsky
\endchange

\bugonpage 3.595 line 3 (00.05.18)
$\displaystyle(uv)^{k\choose2} (-u^{-n}v^{1-n})^k$ \becomes
$\displaystyle(uv)^{j\choose2} (-u^{-n}v^{1-n})^j$
\endchange

\bugonpage 3.595 line 6 of answer 23 (99.06.29)
$(q_n^{h_1+k_1}p_n)$ \becomes $(q_n^{h_1+k_n}p_n)$
\endchange

\bugonpage 3.596 line 3 of answer 27 (00.01.03)
$\displaystyle\sum_n{H_n(w,z)\over(1-z)\ldots(1-z^n)}$ \becomes
$\displaystyle{H_n(w,z)\over(1-z)\ldots(1-z^n)}$
\endchange

\amendpage 3.597 replacement for last paragraph of answer 28 (02.11.23)
\indent The average total displacement of a random permutation is
$(n^2-1)/3$; see exercise 5.2.1--7.
The generating function for total displacement does not appear to
have a simple form.
\em References: C.~Spearman, {\sl British J. Psychology\/ \bf2} (1906),
89--108; P.~Diaconis and R.~L. Graham, {\sl J. Royal Stat.\ Soc.\ \bf
B39} (1977), 262--268.
\endchange

\bugonpage 3.598 line 2 of answer 11 (06.01.01)
if and only if \dots\ and that \becomes
if and only if we have
$\sigma_{p(1)}\interc \cdots\interc
\sigma_{p(t)}=\sigma_1\?\interc\cdots\interc\sigma_t$ and
$p(i)<p(j)$ whenever $\sigma_{p(i)}=\sigma_{p(j)}$ for $i<j$.
We also want to show that,
\endchange

\bugonpage 3.599 answer 14 (02.11.01)
line 11: $(l\cdot a,\,m\cdot b,\,n\cdot c)$ \becomes
$\{l\cdot a,\,m\cdot b,\,n\cdot c\}$\nl
line 14: $(n_1\cdot x_1,\ldots,n_m\cdot x_m)$ \becomes
$\{n_1\cdot x_1,\ldots,n_m\cdot x_m\}$
\endchange

\amendpage 3.599 line 1 of answer 15 (06.01.01)
Theorem 2.3.4.2D \becomes Theorem 2.3.4.2D and Lemma 2.3.4.2E
\endchange

\bugonpage 3.602 last line of answer 2 (06.01.01)
$m=n-q$ and $n-q-1$ \becomes $q=n-m$ and $q=n-m+1$.
\endchange

\bugonpage 3.603 line 1 of answer 11 (99.08.09)
${\sum}{}_{t_1\ge 1,\ldots,t_k\ge 1}$ \becomes
${\sum}{}_{t_1\ge 1,\ldots,t_{k-1}\ge 1}$
\endchange

\bugonpage 3.603 replacements for lines 6, 7, and 9 of answer 11 (06.01.01)
$$\jot5pt
\eqalign{E_1(n)&=1/(n+1)!-1/n!\,;\qquad\qquad E_2(n)=\[n>0]/(n+1)!\,;\cr
E_k(n)&=(-1)^k\sum_{m\ge 0}\,\Bigl({m\atop k-3}\Bigr)\,{\[n>0]\over (n+2+m)!},\qquad k\ge 3.}$$
$$E(z,x)=\sum_{n,k}E_{k+1}(n)z^nx^k=
{ez^2x^2-e^x(1@{-}@x@{+}@zx)(z@{+}@x@{-}@1)-e^{z+x}(1@{-}@z)^2(1@{-}@x)^2\over
e^x z@(z@{+}@x@{-}@1)(1@{-}@x)^2}.$$
\endchange

\amendpage 3.604 answer 16 (02.08.14)
[change the notation from $n\parts k$ to
$\rlap{$\bigr\rangle$}{n\euler k}\llap{$\bigl\langle$}$, because I'm
now using the former notation for partition numbers in Volume~4]
\endchange

\amendpage 3.605 line 9 (02.08.22)
$\langle\langle{n\atop k}\rangle\rangle$ \becomes
$\rlap{$\bigr\rangle$}\bigl\langle{n\atop k}\bigr\rangle\llap{$\bigl\langle$}$
\endchange

\improvepage 3.605 near the top (02.08.22)
line 8: {\sl Paris\/ \bf81} (1875) \becomes {\bf81} (Paris, 1875)\nl
line 9: {\sl Paris\/ \bf97} (1883) \becomes {\bf97} (Paris, 1883)\nl
line 10: (Paris: 1884) \becomes (1884)
\endchange

\bugonpage 3.605 line $-7$ of answer 16 (06.01.01)
$g_n(-1)=0$ \becomes $G_n(-1)=0$
\endchange

\amendpage 3.605 replacement for lines 4 and 5 of answer 19 (03.11.25)
[{\sl Duke Math.\ J. \bf13} (1946), 259--268.
Sections 2.3 and 2.4 of Richard Stanley's {\sl Enumerative Combinatorics\/
\bf1} (1986) discuss rook placement in general.]
\endchange

\bugonpage 3.606 line $-6$ of answer 23 (06.01.01)
$\bigl( u^m+(1-u)^m +\vert u-x\vert^m)/m!$ \becomes
$\bigl( u^m+(1-u)^m -\vert u-x\vert^m\bigr)/m!$
\endchange

\amendpage 3.606 last line of answer 25 (05.02.12)
722]. \becomes 722]; see also W.~Meyer and R.~von Randow, {\sl
Math.\ Annalen\ \bf193} (1971), 315--321.
\endchange

\amendpage 3.607 new answer (00.05.26)
\ans28. The poles of $L(z)$ are the values of $T(1/e)$, where $T(z)$ is
the (multivalued) tree function defined by $T(z)=ze^{T(z)}$.
Thus for $m>0$ we have the convergent series
$$
z_m=-\sigma_m+\sum_{n\ge0}{1\over\sigma_m^{@@n}}\sum_k(-1)^k\Bigl[{n\atop k}\Bigr]
{(\ln\sigma_m)^{n+1-k}\over(n+1-k)!}\,,\qquad \sigma_m=-1-(2m+1)@\pi i
$$
[Corless, Gonnet, Hare, Jeffrey, and Knuth, {\sl Advances in
Computational Mathematics\ \bf5} (1996), 329--359, formula (4.18)]; in
particular, we have $z_m=(2m+{1\over2})\pi i+\ln(2\pi em)+
\bigl({1\over4}-{i\over2\pi}\ln(2\pi em)\bigr)/m+
O\bigl((\log m)^2\!/m^2\bigr)$.\par
Let $P(z)=\sum_{m=0}^\infty\bigl(z/(z-z_m)+z/(z-\bar z_m)\bigr)$. It follows
that $P(x)-P(-x)=\sum_{m=0}^\infty
4\Re\bigl(xz_m/(x^2-z_m^2)\bigr)
=\sum_{m=1}^\infty O\bigl((x\log m)/(x^2+m^2)\bigr)
=\sum_{m=1}^x O\bigl((x\log x)/x^2\bigr)
+\sum_{m=x+1}^\infty O\bigl((x\log m)/m^2\bigr)=O(\log x)$ for $x>1$.
But we know that $L(x)+P(x)=cx$ for some~$c$; hence $2cx=L(x)-L(-x)+O(\log x)$,
and by letting $x\to\infty$ in~\eq(25) we find $c=-1/2$. Hence
$L_1=\sum_{m=0}^\infty2r_m\_\cos \theta_m-1/2$. (This result is due to
Svante Janson.){\advance\baselineskip1pt\par}
\endchange

\amendpage 3.607 another new answer (05.01.29)
\ans29. (a) If $a_1\ldots a_n$ has $2k$ alternating runs and $k$ peaks,
$(n{+}1{-}a_1)\ldots(n+1-a_n)$ has $k-1$ peaks.
(b,c) See L.~W. Shapiro, W.-J. Woan, and S.~Getu, {\sl SIAM J.
Algebraic and Discrete Methods\/ \bf4} (1983), 459--466.
\endchange

\improvepage 3.609 lines 13 and 14 (02.08.18)
[Change the notation from `$a_m$' to the more modern `$e_m$'.]
\endchange

\bugonpage 3.610 answer 23 (06.01.01)
line 4: row $k$ \becomes row $k$ from the bottom\nl
line 8: $\displaystyle \sum_{n\ge1}$ \becomes $\displaystyle\sum_{n\ge0}$
\endchange

\amendpage 3.610 and 611, in answer 23 (07.08.20)
The coefficients $A_{2n}$ are therefore \dots~tangent numbers and
Euler numbers \becomes\nl
The coefficients $E_n$ are
called {\it Euler numbers\/}; with odd index, $E_{2n-1}$ is the
tangent number $T_{2n-1}=(-1)^{n-1}4^n(4^n-1)B_{2n}/(2n)$.
Tables of these numbers appear in {\sl Math.\
Comp.\ \bf 21} (1967), 663--688; the sequence begins
$(E_0,E_1,E_2,\ldots\,)=(1,1,1,2,5,\allowbreak16,61,272,1385,7936,\ldots\,)$.
The easiest way to compute Euler numbers
\endchange

\amendpage 3.611 last two lines of answer 23 (07.07.13)
[A. J. Kempner \dots~349] \becomes
[L.~Seidel, {\sl Sitzungsberichte math.-phys.\ Classe
Akademie Wissen.\ M\"unchen\/ \bf7} (1877), 157--187]
\endchange

\amendpage 3.611 lines 3--5 of answer 28 (00.03.23)
[M. Talagrand \dots~$\Theta(n^{1/6})$.$@$] \becomes\nl
[J.~Baik, P.~Deift, and K.~Johansson, {\sl J.~Amer.\ Math.\ Soc.\
\bf12} (1999), 1119--1178, showed that the
standard deviation is $\Theta(n^{1/6})$; moreover, the probability that the
length is less than $2\sqrt n+tn^{1/6}$ approaches $\exp\bigl(-\int_t^\infty
(x-t)@u^2(x)\,dx\bigr)$, where $u''(x)=2u^3(x)+xu(x)$ and $u(x)$ is
asymptotic to the Airy function ${\rm Ai}(x)$ as $x\to\infty$.]
\endchange

\bugonpage 3.611 line 3 of answer 29 (00.10.28)
$\le \sqrt{n}/e$.\ \ \becomes \ $\le \sqrt{n}/e$.)
\endchange

\bugonpage 3.612 replacement for line $-2$ of answer 33 (06.01.01)
\line{$\ldots,a+1,a,k-1,\ldots,1,0)/\Delta(k+l,\ldots,1,0)$.
The value of $\Delta(k,\ldots,1,0)=k!\ldots2!\,1!$}
\endchange

\bugonpage 3.612 replacement for line 5 of answer 34 (06.01.01)
\line{the shape. If $y_{a+1}=y_a$, the cell $(x_a,y_1)$ has
 a hook of length~$a$; otherwise $(x_{a+b},y_{a+1})$}
\endchange

\bugonpage 3.612 line 12 of answer 35 (06.01.01)
\vskip-3pt
\algindentset{\hskip\parindent H4}
\algstep H4. [Move down or left.] \becomes\smallskip
\algstep H4. [Increase $p$.] Increase $p_{@lk}$ by 1.
\algstep H5. [Move down or left.]
\endchange

\amendpage 3.613 line 4 of answer 37 (02.08.19)
{\bf211} \becomes {\bf A211}
\endchange

\bugonpage 3.613 line 1 of answer 39 (06.01.01)
original permutations \becomes original permutation
\endchange

\bugonpage 3.614 line 5 of answer 41 (06.01.01)
$n+($length \becomes $n-($length
\endchange

\amendpage 3.614 line $-8$ (01.08.14)
{\sl Lecture Notes in Comp.\ Sci.\ \bf807} (1994), 307--325 \becomes
{\sl Algorithmica\/ \bf13} (1995), 180--210
\endchange

\amendpage 3.615 near the top (00.08.01)
line 2: {\sl STOC\/ \bf27} (1995), 178--189 \becomes
{\sl JACM\/ \bf46} (1999), 1--27\nl
lines 3--4: {\sl SODA\/ \bf8} (1997), 344--351 \becomes
{\sl SICOMP\/ \bf29} (1999), 880--892
\endchange

\amendpage 3.619 new sentence for end of answer 7 (02.11.23)
Incidentally, the {\it variance\/} of the stated sum can be shown to equal
$\[n>1]@(2n^2+7)\allowbreak(n+1)/45$.
\endchange

\bugonpage 3.620 throughout answer 10 (01.03.08)
change the local symbols
\.{3F}, \.{3H}, \.{8F}, \.{4H}, \.{5H}, \.{6H}, \.{7F}, \.{5B}, \.{7H},
respectively to
\.{0F}, \.{0H}, \.{7F}, \.{3H}, \.{4H}, \.{5H}, \.{6F}, \.{4B}, \.{6H}.
\endchange

\improvepage 3.620 line 12 (98.08.02)
$NT-S-C$ \becomes $NT-S-C$\qquad\slug
\endchange

\bugonpage 3.620 line 2 of answer 15 (02.11.01)
$g_{n-1}$ \becomes $g_{n-1}(z)$
\endchange

\amendpage 3.623 replacement for lines 7 and 8 of answer 29 (03.12.08)
On the other hand, Marcin Ciura's experiments [{\sl Lecture Notes in
Comp.\ Sci.\ \bf2138} (2001), 106--117] indicate that the
minimum 7-pass $B_{\rm ave}$ ($\approx6879$) is obtained with increments
229 96 41 19 10 4 1, while the sequence
737 176 69 27 10 4 1 yields the smallest total sorting time
($\approx125077u$).
\endchange

\improvepage 3.624 line 22 of answer 31 (98.08.15)
[center the `$T$' in the frequency column]
\endchange

\bugonpage 3.624 line 23 of answer 31 (98.08.15)
is the \becomes is
\endchange

\improvepage 3.625 line $-2$ of answer 33 (98.08.02)
$N-1$ \becomes $N-1$\qquad\slug
\endchange

\improvepage 3.625 bottom line (98.08.02)
$M$ \becomes $M$\quad\slug
\endchange

\bugonpage 3.626 last line of answer 36 (98.08.15)
slow!) \becomes slow!
\endchange

\bugonpage 3.626 replacement for line 13 of answer 37 (00.05.22)
$$\sum_{N\ge0}g''_{NM}(1){M^N w^N\over N!}
=M(M-1)@e^{(M-2)w}\left({w^2\over4}e^w\right)^{\!2}+
 Me^{(M-1)w}\left({w^4\over16}+{5w^3\over18}\right)e^w.$$
[Also change $g'_{MN}(1)$ \becomes $g'_{NM}(1)$ on line 11.]
\endchange

\bugonpage 3.626 line 2 of answer 38 (00.05.22)
converges to \becomes is asymptotic to
\endchange

\bugonpage 3.627 answer 41 (00.10.28)
line 1: We have \becomes (a) We have\nl
line 2: $\rho\losup{k+1}\?/(k+1)-\rho\losup k\?/k$ \becomes
        $(\rho\losup{k+1}\?/(k+1)-\rho\losup k\?/k)/\!@\ln\rho$\nl
line 6: $(k-1)^2$ \becomes $(k-2)^2$\nl
line 8: $(\log\log N)^{-3}$ \becomes $(\log\log N)^{-2}$
\endchange

\bugonpage 3.627 line 10 of answer 42 (00.05.22)
$N/gh$ \becomes $N/gh+1$
\endchange

\amendpage 3.627 bottom line (01.04.11)
307.] \becomes 307.]
However, with a decent sequence of increments the inner loop is not
performed often enough to make this change desirable.\par
\endchange

\amendpage 3.628 line 1 of answer 2 (01.08.16)
{\sl Philos.\ Mag.\ \bf 34} \becomes
{\sl Philosophical Mag.\ \rm(3) \bf 34}
\endchange

\bugonpage 3.629 line 9 of answer 12 (00.08.22)
{\it 05\/} \becomes {\it 06}
\endchange

\improvepage 3.630 line 2 (98.08.02)
$p\ne0$. \becomes $p\ne0$.\quad\slug
\endchange

\bugonpage 3.632 last line of answer 23 (00.06.05)
the form \eq(20) \becomes the form \eq(19)
\endchange

\bugonpage 3.634 line $-7$ of answer 32 (05.10.04)
$H_j+H_{n+1-k}$ \becomes $H_k+H_{n+1-j}$
\endchange

\bugonpage 3.635 last line of answer 38 (00.09.14)
$X_N=\half(A_N-L_N)$ \becomes $X_N=\half A_N$
\endchange

\bugonpage 3.636 last line of answer 41 (99.08.15)
Chapter 11 \becomes Chapter 14
\endchange

\improvepage 3.636 lines 7 and 8 of answer 44 (99.08.15)
[insert a bit of space between these lines]
\endchange

\bugonpage 3.636 line 5 of answer 48 (00.06.05)
${}-\delta_0(n)$ \becomes $-\delta_0(n)+O(n^{-100})$
\endchange

\bugonpage 3.636 replacement for answer 49 (00.06.05)
\ans49.  The right-hand side of Eq.~\eq(40) can be improved to the estimate
$e^{-x}\bigl(1-\half x^2\!/n+O\bigl((x^3{+}x^4)n^{-2}\bigr)\bigr)$.
The effect is to subtract half the sum in exercise 47,
replacing $O(1)$ in \eq(50) by
${2-\half \bigl( 1/\?\ln 2+\delta_1(n)\bigr) +O(n\_)}$. \bigparen(The
``2'' comes from the ``$2/n$'' in \eq(46).\bigparen)
\endchange

\bugonpage 3.637 lines 3 and 4 (00.06.05)
.0000001725, 00041227, \dots, .341 \becomes\nl
.000000172501, .000041227, .0002963, .0008501433, .0062704,
.06797, .1525, .348
\endchange

\bugonpage 3.637 line 2 of answer 53 (99.08.15)
$(p^kq^{n-k}+q^kp^{n-k})@x_n$ \becomes $(p^kq^{n-k}+q^kp^{n-k})@x_k$
\endchange

\improvepage 3.638 last line of answer 54 (99.08.15)
{\sl \"Uber\/} \becomes {\sl \"uber\/}
\endchange

\bugonpage 3.638 line $-10$ of answer 55 (99.08.04)
$R_{i+1}$ \becomes $R_{@l+1}$
\endchange

\bugonpage 3.638 last two lines of answer 55 (99.08.05)
does not look at \dots fast. \becomes
does not look at $K_{N+1}$, but it still might examine $K_0$ in step Q9.
\endchange

\amendpage 3.641 line 3 of answer 15 (06.04.22)
{\bf Step 1.} \becomes {\bf P1.} [Initialize.]
\endchange

\bugonpage 3.641 line $-2$ (02.01.01)
the smallest prime divisor \becomes a prime divisor
\endchange

\bugonpage 3.642 line 1 (06.04.22)
the queue element \becomes a queue element
\endchange

\amendpage 3.642 opening lines (06.04.22)
line 1: {\bf Step 2.} \becomes {\bf P2.} [Advance $q$.]\nl
line 4: {\bf Step 3.} \becomes {\bf P3.} [Check for prime $n$.]\nl
line 6: {\bf Step 4.} \becomes {\bf P4.} [Check for prime $\sqrt n$.]\nl
line 9: {\bf Step 5.} \becomes {\bf P5.} [Advance $n$.]
\endchange

\bugonpage 3.642 line 9 (98.08.02)
return to (b). \becomes return to P2.\quad\slug\nl
\endchange

\bugonpage 3.642 line 16 (01.08.06)
$O(N\log N)$ \becomes $O(N\log N\log\log N)$
\endchange

\amendpage 3.642 last line of answer 15 (06.10.18)
Section 7.1 \becomes Section 7.1.3
\endchange

\amendpage 3.642 answer 16 (06.04.22)
line 1: {\bf Step 1.} \becomes {\bf I1.} [Make a new leaf $j$.]\nl
line 2: {\bf Step 2.} \becomes {\bf I2.} [Find parent of $j$.]\nl
line 3: {\bf Step 3.} \becomes {\bf I3.} [Done?]\nl
line 4: {\bf Step 4.} \becomes {\bf I4.} [Sift and move $j$ up.]\nl
line 4: return to step 2. \becomes return to I2.\quad\slug
\endchange

\bugonpage 3.643 line 1 (06.04.22)
$13N\lg N+O(N)$ \becomes $(13N\lg N+O(N))u$
\endchange

\bugonpage 3.643 line 3 of answer 21 (02.12.23)
$\sum_{0\le q<k}2^q$ \becomes $\sum_{0\le q\le k}2^q$
\endchange

\bugonpage 3.643 line $-2$ of answer 23 (00.09.14)
$\lfloor N/2\rfloor$ \becomes $\lceil N/2\rceil-1$
\endchange

\bugonpage 3.643 last line of answer 25 (00.09.14)
$(2n-3)$ \becomes $(2n-1)$
\endchange

\bugonpage 3.644 line $-7$ (02.12.23)
$\displaystyle\sum_{j=0}^{t-1}$ \becomes $\displaystyle\sum_{j=0}^{t-2}$
\endchange

\bugonpage 3.644 line $-2$ (99.08.15)
$h_n\_$ \becomes $h_N\_$
\endchange

\bugonpage 3.648 line 5 of answer 15 (05.12.04)
twelve \becomes eighteen
\endchange

\bugonpage 3.648 line 6 of answer 15 (03.07.17)
$(p,q,r)$ \becomes $(p,q,s)$
\endchange

\improvepage 3.648 last line of answer 15 (06.02.17)
$8N\lg N+O(N)$ \becomes $(8N\lg N+O(N))u$
\endchange

\improvepage 3.650 line 12 of answer 5 (98.08.02)
\.{DEC1}\quad\.1 \becomes \.{DEC1}\quad\.1\hskip7em\slug
\endchange

\bugonpage 3.652 replacement for line 4 of answer 18 (99.10.05)
\noindent
$\half\?\sum_{k=0}^{CN-1}
\sum_j{N\choose j}p_k^{@j}(1-\nobreak p_k)^{N-j}{j\choose2}=
\half\?\sum_{k=0}^{CN-1}{N\choose2}@p_k^2\le{N-1\over4}\sum_{k=0}^{CN-1}p_kB/C\?$,
because $p_k\le\nobreak B/C\?N\?$.
\endchange

\bugonpage 3.653 line 3 of answer 3 (99.10.05)
Eq.\ 1.2.9--\eq(10).) \becomes Eq.\ 1.2.9--\eq(10).
\endchange

\amendpage 3.653 line 13 of answer 3 (01.08.16)
{\sl Phil.\ Mag.\ \bf 18} \becomes {\sl Phil.\ Mag.\ \rm(4) \bf 18}
\endchange

\improvepage 3.655 line 2 of answer 7 (99.10.05)
6 \becomes $2\cdot3$
\endchange

\bugonpage 3.657 line $-5$ (99.10.05)
noninteger, \becomes noninteger.
\endchange

\amendpage 3.658 answer 35 (02.08.20)
[delete this answer, as the exercise has changed]
\endchange

\bugonpage 3.658 answer 1 (98.07.09)
$S(n)+S(n)$ \becomes $S(m)+S(n)$
\endchange

\amendpage 3.664 lines 2--4 of answer 23 (02.02.09)
D. Dor \dots\ proved \becomes
D.~Dor and U.~Zwick
have shown that the actual lower limit is strictly greater
than~2, while the upper limit is less than 2.942
[{\sl SICOMP\/ \bf28} (1999), 1722--1758;
 {\sl SIAM J. Disc.~Math.\ \bf14} (2001), 312--325].
They also have proved
\endchange

\amendpage 3.666 new sentence for answer 7 (02.06.16)
(Notice that the modified network has delay~8.)
\endchange

\amendpage 3.668 new answer (03.09.12)
\ans20. \hfil\vtop{\kern-6pt\epsfbox{\figdir/v3.668}\vskip-100pt}%
{\parfillskip=0pt\par}\nobreak\vskip-\baselineskip\vskip-\parskip
\hangindent-1.8in\hangafter-9\noindent
\indent (a) First note that $\hat V_3(n)\ge \hat V_3(n-1)+2$ when $n\ge 4$:
By symmetry the first comparator may be assumed to be $[1:n]$;
after this must come 
a network to select the third largest of $\langle x_2,x_3,\ldots,x_n\rangle$,
and another comparator touching line 1. On the other hand, $\hat V_3(5)\le 7$,
since four comparators find the min and max of $\{x_1,x_2,x_3,x_4\}$,
then we sort the other three.\hfil\break
\indent(b) A subtle construction by M.~W. Green, shown for $n=11$, does
the job. (Equality probably holds.)\looseness=-1
\endchange

%\amendpage 3.669 last line of answer 31 (04.05.27)
%argument. \becomes argument; J.~Kahn has found a simpler
%proof [{\sl Proc.\ Amer.\ Math.\ Soc.\ \bf130} (2002), 371--378].
%\endchange

\amendpage 3.669 last three lines of answer 31 (05.11.01)
No simple \dots\ \becomes
The asymptotic formula
$\delta_{2m}=\exp\bigl({2m-1\choose\raise1pt\hbox{$\scriptstyle m$}}\ln2+{2m\choose m+1}/2^{m+1}+
{1\over16}(m-1)\sqrt{m/\pi}+O(m^{-1/2})\bigr)$ has been established by
A.~D. Korshunov and A.~A. Sapozhenko, with a similar formula~for
$\delta_{2m+1}$; see
{\sl Russian Math.\ Surveys\/ \bf58} (2003), 929--1001, Theorem~1.8.
\endchange

\amendpage 3.669 notation change in answer 32 (05.08.08)
line 1: $\theta\le\psi$ \becomes $\theta\subseteq\psi$\nl
line 3: ``$\le$'' \becomes ``$\subseteq$''
\endchange

\bugonpage 3.669 line 4 of answer 32 (99.10.05)
$G_T$ \becomes $G_t$
\endchange

\improvepage 3.669 last line of answer 32 (05.08.08)
$z[(x_1\ldots x_t)_2]$ \becomes $z_{(x_1\ldots x_t)_2}$
\endchange

\bugonpage 3.669 answer 35 (99.10.05)
line 4: and$[n@{-}@1{:}n]$ \becomes and $[n@{-}@1{:}n]$\nl
line 5: $(k-1)D_{k-1}$ \becomes $(k-1)D_{k+1}$
\endchange

\amendpage 3.670 last line of answer 37 (01.01.28)
{\sl Amusements\/} \becomes
{\sl Strand\/ \bf46} (1913), 352, 472; {\sl Amusements}
\endchange

\improvepage 3.670 line $-11$ (02.08.22)
{\sl Paris\/ \rm(I) \bf295} (1982) \becomes (I) {\bf295} (Paris, 1982)
\endchange

\bugonpage 3.671 line 3 of answer 40 (98.10.27)
$t=4n$ \becomes $t=4n+\sqrt n\,\ln n$
\endchange

\bugonpage 3.671 replacement for lines 4--7 of answer 40 (98.10.28)
\indent
Experiments show that the expected time to reach {\it any\/} primitive sorting
network\dash---not necessarily the bubble sort\dash---is very nearly~$2n^2$.
Curiously, R.~P. Stanley and S.~V. Fomin have proved that if the
comparators $[i_k:i_k{+}1]$ are chosen nonuniformly in such a way that $i_k=j$
occurs with probability $j/{n\choose2}$, the corresponding expected time comes
to exactly ${n\choose2}H_{n\choose2}$.
\endchange

\bugonpage 3.671 line 1 of answer 45 (05.02.03)
at most $2^{@t}$ \becomes at most $2^{@l}$.
\endchange

\bugonpage 3.671 lines 7 and 8 of answer 46 (05.02.03)
either $y_{k+1}\le\cdots\le y_n$ or \dots\
at least $\min(n{-}k{+}1,k)\ge\min(n{-}l{+}1,l{+}m)$\nl
\becomes
either $y_{k+1}\le\cdots\le y_n$ and $k\le l$ or $y_1\le\cdots\le y_{k-1}$
and $k\ge l+m$; so at least $\min(n{-}l{+}1,l{+}m)$
\endchange

\bugonpage 3.671 line 4 of answer 49 (99.10.05)
$\{0,0,0,1,1,1\}$) \becomes $\{0,0,0,1,1,1\}$
\endchange

\improvepage 3.672 line $-11$ (99.05.11)
consists of two \becomes consist of two
\endchange

\amendpage 3.676 lines 1 and 2 of answer 66 (05.02.14)
no pair sequence \dots\ 10 comparisons. \becomes\nl
every pair sequence beginning with
$(1,2)(2,3)(3,4)(4,5)(1,5)$ will avoid at least one subsequent comparison.
\endchange

\amendpage 3.676 replacement for answer 67 (05.02.14)
\ans67. Suppose $c_i=j$ for exactly $t_j$ values of $i$. For the restricted case
we need to prove that $\sum_j t_j/(2+j)$ is minimized when $(t_0,t_1,\ldots,
t_{N-2})=(N{-}1,\ldots,2,1)$. Gil Kalai has shown that the achievable
vectors $(t_0,t_1,\ldots,t_{N-2})$ are always lexicographically
$\ge(N{-}1,\ldots,2,1)$; see {\sl Graphs and Combinatorics\/
\bf1} (1985), 65--79.
\endchange

\improvepage 3.690 line 5 of answer 23 (99.11.14)
$u^{(1)}$, \dots$@$. \becomes $u^{(1)}$, \dots,~$u^{(q-1)}$.
\endchange

\bugonpage 3.681 line 4 of answer 6 (03.01.10)
$\alpha p(\alpha\losup{-\?1}q''(\alpha\losup{-\?1})/
q'(\alpha\losup{-\?1})\losup3\!@$, \becomes
$\alpha p(\alpha\losup{-\?1})@q''(\alpha\losup{-\?1})/
q'(\alpha\losup{-\?1})\losup3\!@$,
\endchange

\improvepage 3.685 line 1 of answer 5 (03.01.10)
$k$ pass \becomes $k$th pass
\endchange

\bugonpage 3.688 line $-9$ (03.01.10)
$3c_n+2d_n+c_n$ \becomes $3c_n+2d_n+e_n$
\endchange

\improvepage 3.692 near the top (03.01.10)
line 1: $D[p]$ \becomes \.{D[$p$]}\nl
line 2: $D[p]$ \becomes \.{D[$p$]}\nl
line 2: $D[q]$ \becomes \.{D[$q$]}
\endchange

\improvepage 3.692 answer 5.4.6--5 (03.01.10)
line 4: $C[i]$ \becomes \.{C[$i$]}\nl
line 5: $C[@j]$ \becomes \.{C[$@j$]}
\endchange

\bugonpage 3.696 line 7 of answer 2 (01.03.15)
$k+1=q_{i(q+1)}$ \becomes $k+1=a_{i(q+1)}$
\endchange

\bugonpage 3.697 line 3 of answer 9 (01.03.16)
$D(\tau)=dE(\tau)$, $E(\tau)=tn+dr$ \becomes
$D({\cal T})=dE({\cal T})$, $E({\cal T})=tn+dr$
\endchange

\amendpage 3.704 line 2 of answer 17 (05.05.23)
Of course \becomes Management Science Research Report~43,
UCLA (1955). Of course
\endchange

\bugonpage 3.704 answer 18 (99.11.14)
first line: $q_j<r_1$ \becomes $q_j<r_j$\nl
last line: 10).] \becomes 10.]
\endchange

\bugonpage 3.708 line 3 of answer 2 (06.01.12)
``$\.{RTAG(P)}\ne0$'' \becomes ``$\.{RTAG(P)}=0$''
\endchange

\bugonpage 3.711 line 8 of answer 32 (01.04.20)
$\max\sum_{k=0}^n q_k$ \becomes $\max_{k=0}^n q_k$
\endchange

\bugonpage 3.712 lines 5 and 10 of answer 40 (98.12.03)
$q_{k-2}(l_{k-2}-l_{k-4})$ \becomes $q_{k-2}(l_{k-4}-l_{k-2})$
\endchange

\bugonpage 3.712 replacement for answer 42 (04.09.18)
\ans42. Let $q_j=\.{WT(P$_j$)}$. Steps C1--C4, which move $q_{k-1}+q_k$
into position between $q_{j-1}$ and~$q_j$, can spoil~\eq(31) only
at the point $i=j-1$.
\endchange

\bugonpage 3.713 in answer 44 (04.09.15)
line 4: $\.{LLINK(X$_j$)}\gets t$ \becomes $\.{LLINK(X$_j$)}\gets \.X_t$\nl
line 6: $\.{RLINK(X$_j$)}\gets t$ \becomes $\.{RLINK(X$_j$)}\gets \.X_t$
\endchange

\amendpage 3.713 lines 6--8 of answer 49 (03.09.17)
Luc Devroye and Bruce Reed \dots, where \becomes
Bruce Reed [{\sl JACM\/ \bf50} (2003), 332]
and Michael Drmota [{\sl JACM\/ \bf50} (2003), 333--374],
who proved that the average
height is $\alpha\ln n-(3\alpha\ln\ln n)/(2\alpha-2)+O(1)$
and the variance is $O(1)$, where
\endchange

\bugonpage 3.715 line 2 of answer 9 (98.07.29)
$\log_{(\sqrt10+2)/3}n$ \becomes $\log_{(\sqrt{10}+2)/3}n$
\endchange

\bugonpage 3.715 last line of answer 11 (99.11.28)
6.2.4--8. \becomes 6.2.4--8.]
\endchange

\amendpage 3.718 line 4 (06.03.23)
33--43.] \becomes
33--43; N.~Blum and K.~Mehlhorn, {\sl Theoretical Comp.\ Sci.\ \bf11}
(1980), 303--320.]
\endchange

\bugonpage 3.722 line 9 (99.11.28)
(1991) \becomes (1992)
\endchange

\improvepage 3.722 bottom line (98.08.02)
rA and rX. \becomes rA and rX.\quad\slug
\endchange

\amendpage 3.726 line 8 of answer 27 (00.10.06)
not hard to prove. \becomes not hard to prove.
Moreover, $\alpha$ turns out to be identical to the
constant defined quite differently in 5.2.3--\eq(19);
see Karl Dilcher, {\sl Discrete Math.\ \bf145} (1995), 83--93.
\endchange

\bugonpage 3.726 line 2 of answer 31 (98.07.29)
W. Prodinger \becomes H. Prodinger
\endchange

\bugonpage 3.727 lines 7 and 8 of answer 34 (99.11.28)
${}+\cdots$ \becomes ${}+\cdots{})$ \qquad (twice)
\endchange

\amendpage 3.728 new answer (99.03.08)
\ans45. The probability of $\{\.{THAT},\.{THE},\.{THIS}\}$ before
$\{\.{BUILT},\.{HOUSE},\.{IS},\.{JACK}\}$,
$\{\.{HOUSE},\allowbreak\.{IS},\allowbreak\.{JACK}\}$ before $\{\.{BUILT}\}$,
$\{\.{HOUSE},\.{IS}\}$ before $\{\.{JACK}\}$,
$\{\.{IS}\}$ before $\{\.{HOUSE}\}$,
$\{\.{THIS}\}$ before $\{\.{THAT},\.{THE}\}$,
and $\{\.{THE}\}$ before $\{\.{THAT}\}$
is ${3\over7}\cdot{3\over4}\cdot{2\over3}\cdot{1\over2}\cdot{1\over3}\cdot
{1\over2}={1\over56}.$
\endchange

\amendpage 3.732 line 4 of answer 29 (99.02.25)
1274.] \becomes 1274;
see also R.~Pyke, {\sl Annals of Math.\ Stat.\ \bf30} (1959), 568--576,
Lemma~1.]
\endchange

\bugonpage 3.733 line 16 (00.12.05)
Conversely, The \becomes Conversely, the
\endchange

\amendpage 3.734 new copy replacing lines 1--5 of answer 32 (02.01.05)
\ans32. Let
$$s_j\;=\;\sum_{k=0}^j\,(b_{k\mod M}-1).$$ % displayed for paging, 02.01.05)
Then, as observed by Svante Janson, we have $c_j=\max_{k\ge j}(s_k-s_j)$,
a quantity that is well defined because $\lim_{k\to\infty}s_k=-\infty$.
\endchange

\bugonpage 3.736 last lines of answer 40 (05.02.13)
When $\alpha=1$ \dots bounded by 2.12. \becomes
When $\alpha =1$ this is about 2.65, so the
standard deviation is bounded by 1.63. [Svante Janson (to appear) has
found the asymptotic moments of all orders, also when the search is successful.]
\endchange

\amendpage 3.740 line 2 of answer 56 (98.05.29)
to appear \becomes 37--71
\endchange

\amendpage 3.742 lines 7 and 8 of answer 68 (98.11.23)
D.~E. Knuth, P.~Flajolet, \dots to appear. \becomes
P.~Flajolet, P.~V. Poblete, and A.~Viola,
{\sl Algorithmica\/ \bf22} (1998), 490--515;
D.~E. Knuth, {\sl Algorithmica\/ \bf22} (1998), 561--568.
\endchange

\amendpage 3.742 line 2 of answer 73 (99.12.19)
characters. \becomes characters.
[It was invented as early as 1970 by Alfred~L. Zobrist, whose original
technical report has been reprinted in
{\sl ICCA Journal\/ \bf13} (1990), 69--73.]
\endchange

\amendpage 3.743 new answer (07.09.26)
\ans78. Let $g(x)=\lfloor x/2^k\rfloor\mod 2^{n-k}$ and
$\delta(x,x')=\sum_{b=0}^{2^k-1}\[g(x+b)=g(x'+b)]$. Then
$\delta(x+1,x'+1)=\delta(x,x')+\[g(x+2^k)=g(x'+2^k)]-\[g(x)=g(x')]
=\delta(x,x')$. Also $\delta(x,0)=(2^k\dotminus(x\mod2^n))
+(2^k\dotminus((-x)\mod2^n))$ when $0<x<2^n$, where $a\dotminus b=\max(a-b,0)$.
Therefore $\delta(x,x')=
\bigl(2^k\dotminus((x-x')\mod2^n)\bigr)+
\bigl(2^k\dotminus((x'-x)\mod2^n)\bigr)$
when $x\not\equiv x'$ (modulo~$2^n$).\par
Now let $A=\{a\mid 0<a<2^n,\ a\ \rm odd\}$ and
$B=\{b\mid 0\le b<2^k\}$. We want to show that
$\sum_{a\in A}\sum_{b\in B}\[g(ax+b)=g(ax'+b)]\le R/M=
2^{n-1+k}\!/2^{n-k}=2^{2k-1}$ when $0\le x<x'<2^{\mathstrut n}$. And indeed, if
$x'-x=2^pq$ with $q$~odd, then we have
$$\displaylines{
\sum_{a\in A}\,\sum_{b\in B}\,\[g(ax+b)=g(ax'+b)]=
\sum_{a\in A}\delta(ax,ax')=
2\sum_{a\in A}\bigl(2^k\dotminus((2^paq)\mod2^n)\bigr)
\hfill\cr\hfill
=2^{p+1}\!\!\sum_{j=0}^{2^{n-p-1}-1}\bigl(2^k\dotminus 2^p(2j+1)\bigr)
=2^{p+1}\!\!\sum_{j=0}^{2^{k-p-1}-1}(2^k-2^p(2j+1))\[p<k]
=2^{2k-1}\[p<k].\cr}$$
[See {\sl Lecture Notes in Computer Science\/ \bf1672} (1999), 262--272.]
\endchange

\amendpage 3.744 in the third paragraph of answer 1 (03.05.04)
{\it Sperner's Lemma} \becomes {\it Sperner's Theorem}\nl
Sperner's Lemma \becomes Sperner's Theorem
\endchange

\bugonpage 3.744 just before the final display in answer 4 (99.12.12)
And If \becomes And if
\endchange

\amendpage 3.745 new material at end of answer 10 (05.12.04)
Kirkman wrote his paper in response to a substantially
more general problem posed by W.~S.~B. Woolhouse, namely
to find the maximum number of $t$-element subsets of $\{1,\ldots,n\}$
in which no $q$-element subset appears more than once; that
problem remains unsolved. [See {\sl Lady's and Gentleman's Diary\/}
(1844), 84; (1845), 63--64; (1846), 76, 78; (1847), 62--67.]
\endchange

\bugonpage 3.747 line 6 of answer 18{(e)} (01.03.25)
$32\times8=512$ \becomes $32\times16=512$
\endchange

\bugonpage 3.747 last line of answer 18 (01.06.04)
also yields on \becomes also yields an
\endchange

\let\defaultpointsize=\tenpoint % begin appendices

\planforpage 3.749 Table 2 (99.05.06)
In the next edition I plan to give these constants to 36 hexadecimal
places, instead of 45 octal places.
\endchange

\amendpage 3.750 line 5 (00.09.16)
6.3--26, and 6.3--27. \becomes and 6.3--26.
\endchange

\bugonpage 3.753 entry for Kronecker delta (98.08.14)
1.2.6 \becomes 1.2.3
\endchange

\bugonpage 3.755 entry for $\exp x$ (99.08.09)
1.2.2 \becomes 1.2.9
\endchange

\bugonpage 3.756 entry for $\Im z$ (01.01.27)
imaginary part $z$ \becomes imaginary part of $z$
\endchange

\expandafter\ifx\csname indexeject\endcsname\relax\else\vfill\eject\fi
\amendpage 3.757 and following (98.02.16)
Miscellaneous changes to the existing index of Volume~3 are collected here,
including corrections and amendments to the old entries as well as new entries
that are occasioned by the new material. Thus, the lines of the full index
that have changed serve also as an index to the present document. However,
when a correction or amendment has caused an old index entry to be deleted,
the deletion is usually not indicated.

\input exotic
\begindoublecolumns
\indexformat
\gdef\Uni1.08:{\bitmap24:1.08:}
\hangindent 2em
$0@$--1 matrices, 660. % 20th
$0@$--1 principle, 223, 224, 245, 667, 668. % 20th
$\zeta(x)$ (number of $0@$s), 235; \also Zeta function. % 20th
$\nu(x)$ (number of 1s), 235, 643, 644, 717. % 20th
$\pi$ (circle ratio), as ``random'' example, 17, 370, 385, 547, 552, 733. % 21st
Agarwal, Ramesh Chandra ({\dn rm\kern-1em\hbox{\char3}\kern0.2emf \char99\char6d\kern-.5em\char'375\kern.5em\ ag\llap{\char125}vAl}), 359,  389. % 19th
Aggarwal, Alok ({\dn aAlok ag\llap{\char125}vAl}), 698. % 19th
Airy, George Biddle, function, 611. %4th
Alternating runs, 46, 607. % 21st
{\sl Anuyogadv\=aras\=utra\/} ({\dn an\kern-.4em\char0\kern.45emyog\char146Ars\kern-.2em\lower.05em\hbox{\char1}\kern.1em\char47}), 23. % 20th
Arora, Sant Ram ({\dn s\char6t rAm arowA}), 455. %4th
Bachrach (= Gilad-Bachrach), Ran ({\heb\Hfc/\Hrr/\Hcc/\Hbb/}-$\!${\heb\Hdd/\Hai/\Hll/\Hgg/ \Hfn/\Hrr/}), 403. % 24th
Baik, Jinho (\kern-1pt\KC
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 0003ffffc03c03c0000003ffff803c0380000003c003803c038000000380038038038000%
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\KC
<000000000000000000000000000000000000000000000000000000000000000000000000%
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 000000001f8001e000000000001f0001e000000000003e0001e000000000007e0001e000%
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 001c0000000001e00000300000000001e00000c00000000001c00000000000000001c000%
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 000000078000008000000000078000000000000000078000000000000000078000000000%
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>%% Unicode char "c9c4 (formerly "3a95)
\KC
<000000000000000000000000000000000000000000000000000000000000000000000000%
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 0000000ffff00000000000003f01f80000000000007c007c000000000000f8001e000000%
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>%% Unicode char "d638 (formerly "3cd2)
), 611. % 19th
Berkovich, Simon Yakovlevich ({\rus Berkovich, Seme0n Yakovlevich}), 496. % 10th
Betz, Bernard Keith, 268, 288. % 8th
Bh\=askara II, \=Ac\=arya, son of M\=ahe\'svara ({\dn BA\char45krA\char99A\hbox{y\kern-0.3em\hbox{\char13}}}, {\dn mAh\kern-.85em{\char3}\char'230rp\llap{\lower.15em\hbox{\char0}\kern.3em}/}), 23. % 21st
Bhattacharya, Kailash Nath ({\bg\C119\C107\C108\C059\\170\ \C110\C059\C113\ \C038\C124\C059\C099\C059\C121\C082\C042}), 746. % 23rd
Binary recurrences, 135, 167, 630, 644, 653. % 9th
Bit reversal, 621. % 19th
Bitner, James Richard, 403, 478, 703. % 7th
Bitwise or, 529, 571. % 19th
Bleier, Robert Edward, 578. % 10th
Bloom, Burton Howard, 572--573, 578, 744. % 11th
Blum, Norbert Karl, 718. % 21st
Boesset, Antoine de, 24. % 17th
Bravais, Auguste, 518. % 13th
Bravais, Louis, 518. % 13th
Buckets, 541--544, 547--548, 555, 564. %5th
Carroll, Lewis (= Dodgson, Charles Lutwidge), 207--208, 216, 584. %4th
Chinese mathematics, 36. % 8th
Chung, Moon Jung (\kern-1pt\KC
<000000000000000000000000000000000000000000000000000000000000000000000000%
 000000000000000000000000000001fc0000000000000001ff00000000000000007fc000%
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 001e000001e00380000038000000c003800000e000000000038000018000000000038000%
 00000000001c01000000000000000f00000000000000000f00000000000000000f000000%
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 0000000078001f000000000000f0000f800000000000f00007800000000001e00003c000%
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 00000001e00003c00000000001f00007c00000000000f00007800000000000f8000f8000%
 000000007c001f0000000000003e003f0000000000001f80fe0000000000000ffff80000%
 0000000003fff0000000000000007f800000000000000000000000000000000000000000%
>%% Unicode char "c815 (formerly "3a41)
\KC
<000000000000000000000000000000000000000000000000000000000000000000000000%
 0000000000000000000000000000000000000000000000000000000001fc000007f00000%
 0003fffffffff8000000007ffffffff8000000001f800001f8000000000f800001f00000%
 00000f800001f0000000000f800001f0000000000f800001e00000000007800001e00000%
 000007800001e00000000007800001e00000000007800001c00000000007800001c00000%
 000007800001c00000000007800001800000000007800001800000000007800001800000%
 000007800fffe00000000007ffffffe00000000007ffffffe00000000007800000000000%
 000007800000000000000007000000000000000003000000000000000000000000000000%
 000000000000000000000000000000000000000000000000003f800d00000000001fffe0%
 07ffffffffffffffe001ffffffffffffffe0007ffc001e00000000000f80001e00000000%
 000000001e00000000000000001e00000000000000001e00000000000000001e00000000%
 000000001e00000000000000001e00000000000000001e00000000000000001e00000000%
 000000001e000000000001f8001e000000000007ff001e000000000001ff801e00000000%
 00007f801c0000000000001f001c0000000000001f001c0000000000001e001c00000000%
 00001e00080000000000001e00000000000000001e00000000000000001e000000000000%
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 000007ffffffff0000000003fffffffe0000000000000000000000000000000000000000%
 000000000000000000000000000000000000000000000000000000000000000000000000%
>%% Unicode char "bb38 (formerly "375b)
\KC
<000000000000000000000000000000000000000000000000000000000000000000000000%
 000000000000000000000000000001fc0000000000000001ff00000000000000007fc000%
 0000000000001fe0000000000000000fe00000000000000003e00000000000000003c000%
 00000000000003c000000000000fc003c000000fffffffc003c0000007ffffffc003c000%
 0000f0001f8003c000000000003f0003c000000000003e0003c000000000007e0003c000%
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 0000000003fff0000000000000007f800000000000000000000000000000000000000000%
>%% Unicode char "c815 (formerly "3a41)
\ =
\JC48:48:0:40% J3702
<0c00060000000f000780000007c007c3000c07e00fc3801e03f00f83ffff01f00e03ffff%
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<0000000380000000000000000001e0000000000000000000f0000000000000000000fc00%
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\Uni1.08:24:24:-1:20% Unicode char "6968
<0e01c00c01800c01840c01fe0c01800d21847fbffe0c300e1c300c1f300c1dbffc3cf00c3cf00c6c300c6c3ffc4c300ccc300c0c300c0c3ffc0c366c0c06300c0c180c180c0c600c>%
), 673. % 19th
Ciura, Marcin Grzegorz, 95, 623. % 14th
Colin, Andrew John Theodore, 453, 454. %4th
Compositions, 286--287. % 11th
Corless, Robert Malcolm, 607. %6th
Deift, Percy Alec, 611. %4th
Derr, John Irving, 547. %13th
Descents of a permutation, 35, 46, 47, 606. % 21st
Diaconis, Persi Warren, 597. % 12th
Diagram of a partial order, 61--62, 183--184, 187. %5th
Digital tree search, 496--498, 517, 546--547. %5th
Dijkstra, Edsger Wybe, 636. % 19th
Dilcher, Karl Heinrich, 726. %7th
Disorder, measures of, 11, 22, 72, 134, 389. % 12th
Displacements, variance of, 556, 619. % 12th
Drmota, Michael, 713. % 15th
El-Yaniv, Ran ({\heb\Hbb/\Hyy/\Hnn/\Hyy/-\Hll/\Haa/ \Hfn/\Hrr/}), 403. %4th
Empirical data, 94--95, 403, 434--435, 468--470. %4th
Euler, Leonhard ({\rus Ei0lerp2, Leonardp2} = {\rus E1i0ler, Leonard}), 8--9, 19--21, 35, 38--39, 395, 593--594, 726. %11th
Eulerian numbers, table, 37. % 16th
Exponential integral, 105, 137, 735. %6th
External searching, 403--408, 481--491, 496, 498--500, 541--544, 549, 555, 562--563, 572--573. % 8th
External path length: Sum of the level numbers \dots. % 16th
Feller, Willibald (= Vilim = Willy = William), 513. %4th
Fibonacci, Leonardo, of Pisa (= Leonardo filio Bonacii Pisano), 424. % 22nd
Free distributive lattice, 239. % 11th
Gassner, Betty Jane, 40--41, 262. % 7th
Gau{\ss} (= Gauss), Johann Friderich Carl\indexbreak(= Carl Friedrich), 395. % 7th
Genetic algorithms, 226, 229. % 9th
Getu, Seyoum ({\et s\\317m g\&261}), 607. % 19th
Gilad-Bachrach, Ran ({\heb\Hfc/\Hrr/\Hcc/\Hbb/}-$\!${\heb\Hdd/\Hai/\Hll/\Hgg/ \Hfn/\Hrr/}), 403. % 24th
Golin, Mordecai Jacob ({\heb\Hfn/\Hyy/\Hll/\Hvv/\Hgg/ \Hbb/\Hqq/\Hai/\Hyy/ \Hyy/\Hcc/\Hdd/\Hrr/\Hmm/}, \GC75:75:-3:62% G2463
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), 649. % 13th, corrected in 14th
Gonnet Haas, Gaston Henry, 489, 533, 565, 607, 707, 734. %6th
Gore, John Kinsey, 385. %6th
Graham, Ronald Lewis (\GC72:74:-4:61% G2480
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 000000000000038000000000000000020000>%% Unicode char "845b
\\\GC75:65:-3:60% G3302
<00000003e0000000000000000001f8000000000000000001fe000000000000000000ff00%
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\JC48:48:0:40% J5581
<00fc0000000000f80000000000f00000006000f0000000f000f0fffffff800f07ffffffc%
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 >%% Unicode char "6046
), 198, 202--203, 205--206, 242, 550, 597, 729, 760. % 12th
Greek mathematics, 420. % 8th
Green, Milton Webster, 227, 239, 667, 668, 673. % 15th
Greniewski, Marek J\'ozef, 513. % 22nd
Guibas, Leonidas John ({\grk Gk'impas, Lewn'idas Iw'annou}), 477, 525, 709, 737. % 24th
Hall, Marshall, Jr\period, 511, 578. % 9th
Hannenhalli, Sridhar Subrahmanyam\indexbreak ({\dn \char128FDr s\kern-0.4em\char0\kern0.5em\lower0.2em\hbox{\char125}\kern-0.8emb\char156\char23y\kern0.5em\lower0.1em\hbox{\char94}\kern-0.5emm h\char6n\kern-0.8em\char3nhSlF}), 615. % 19th
Hare, David Edwin George, 607. %6th
Heap, $t$-ary, 644. % 21st
Hindenburg, Carl Friedrich, 14. %9th
Hindu mathematics, 23. % 14th
Hwang, Frank Kwangming (\GC74:74:-4:61% G2738 edited by hand!
<%
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7fffffffffffffffffc0%
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 3e00000000000001c00020000000000000004000>%% Unicode char "9ec3
\GC74:71:-3:60% G2566
<000000001c0000000000000000001f8000000000000000001fe000000000000000001fe0%
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 001fffffffc0ffe00000000fffffff00ff800000000000000000>%% Unicode char "5149
\GC63:72:-9:59% G3587
<00000000700000e0000000007c0003f0000000007f0007f8000000007ffffffc00000e00%
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 >%% Unicode char "660e
), 188, 195, 202--206. % 7th
IBM Corporation, 8, 169, 193, 316, 385, 489, 547. % 21st
{\it in situ\/} permutation, 79--80, 178. % 8th
Indian mathematics, 23. % 8th
Isaac, Earl Judson, 99. % 23rd
Jackson, James Richard, 407. % 20th
Janson, Carl Svante, 607, 627, 734, 736. % 19th
Japanese mathematics, 36. %8th
Jeffrey, David John, 607. %6th
Jewish mathematics, 23. % 21st
Jiang Ling (\GC79:75:-1:60% G2913
<00e0000000000000000001f8000000000000000000ff0000000000000000007fe0000000%
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\GC77:78:-2:63% G3375
<000380000001c00000000003e0000001f80000000003f0000003fe0000000003fc000003%
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 07e0000000000000000003e0000000000000000003000000>%% Unicode char "5cad
), 589. % 23rd
Johansson, Kurt Ove, 611. %4th
%Kahn, Jeffry Ned, 669. % 17th, removed in 21st
Kaplansky, Irving, 46. % 16th
Karpi\'nski (= Karpinski), Marek Mieczys{\l}aw, 454. % 21st
Kirschenhofer, Peter, 576, 634, 644, 726. %5th
Kleitman, Daniel J (Isaiah Solomon), 452, 454, 744. % 21st
Knuth, Donald Ervin, \dots, 604, 607, 627, \dots. %6th
Korshunov, Aleksey Dmitrievich ({\rus Korshunov, Aleksei0 Dmitrievich}), 669. % 21st
Kwan, Lun Cheung, 657. % 19th
Larmore, Lawrence Louis, 454. % 21st
Lattice paths, 86--87, 102--103, 112--113, 134, 579. %5th
Li Shan-Lan (\GC72:74:-5:61% G3278
<00000000f00000000000000000fc0000000000000000fe0000000000000000ff80000000%
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 00000000fc0000000000000000f000000000>%% Unicode char "674e
\GC74:73:-3:61% G4138
<000007000003e000000000000fc00003f8000000000007f80007f8000000000003fc0007%
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 0001f80000000fc00000>%% Unicode char "5584
\JC47:48:-1:40% J4586
<0001801800000001e01e00000001f01f00180001e01e003cfffffffffffefffffffffffe%
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 >%% Unicode char "862d
), 36. %8th
Liang, Franklin Mark, 722, 729. % 13th
Liddell Hargreaves, Alice Pleasance, 584. %4th
Linial, Nathan
 ({\heb\Hll/\Haa/\Hyy/\Hnn/\Hyy/\Hll/ \Hfn/\Htv/\Hnn/}), 660. % 10th
Lipski, Witold, Jr\period: delete this name. %4th
Litwin, Samuel, 578. %4th
Litwin, Witold Andr\'e, 548--549. %4th
Luke, Richard C\period, 547. %13th
Majorization, 406. % 20th
Markov (= Markoff), Andrei Andreevich ({\rus Markov, Andrei0 Andreevich}), the elder, process, 340--341. % 19th
Martzloff, Jean-Claude, 36. % 8th
Matsunaga, Yoshisuke (\JC48:48:0:40% J3030
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\JC48:47:0:39% J1742
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\JC46:48:-1:40% J4641
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 >%% Unicode char "826f
\JC48:48:0:40% J4111
<001c000c000c003c001e001efffcffff3fff7ffc7fff1fff003c01e0003c003c01e0003c%
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 >%% Unicode char "5f3c
), 36. %8th
Mersenne, Marin, 23--24. % 17th
Meyer, Werner, 606. % 19th
McAndrew, Michael Harry, 502. % 11th
M\"obius, August Ferdinand,\indexnoperiod
\sub function $\mu(\pi)$, 33. %4th
Monomial symmetric function, 609. % 11th
Monotone Boolean functions, 239. % 21st
%Monotone Boolean formula, 239. % 11th
Mooers, Calvin Northrup, 571. % 7th
Naor, Simeon (= Moni;\indexbreak {\heb\Hrr/\Hvv/\Haa/\Hnn/ \Hyy/\Hnn/\Hvv/\Hmm/}, {\heb\Hfn/\Hvv/\Hai/\Hmm/\Hsh/}), 708. % 10th
N\=ar\=aya\d{n}a Pa\d{n}\d{d}ita, \vadjust{\vskip1pt}son of N\d{r}si\:mha\indexbreak({\dn nArAyZ pE\char23Xt}, {\dn n\llap{\lower.12em\hbox{\char2}\kern.4em}Es\llap{\char92\kern-.05em}h-y p\llap{\lower.15em\hbox{\char0}\kern.3em}/,}), 270. % 19th
O'Connor, Daniel John, 225. %4th
Odd-even merge, 223--226, 228, 230, 243, 244. % 10th
Odlyzko, Andrew Michael, 630, 715. %5th
Okoma, Seiichi (\JC48:48:0:40% J3471
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\JC48:4:0:20% J1676
<00000000000c00000000001effffffffffff7fffffffffff>%% Unicode char "4e00
), 644. %5th
\.{OR} (bitwise or), 529, 571. % 19th
Oriented trees, 47, 71, 599. % 21st
Page, Ralph Eugene, 385. % 23rd
Partitions of a set, 605, 653. % 21st
Patents, vi, 225, 231, 244, 255, 312, 315--316, 369, 384--385, 394, 675, 729. % 22nd
Paths on a grid, 86--87, 102--103, 112--113, 134, 579. % 11th
Patterns in permutations, 61. % 19th
Peaks of a permutation, 47, 604. % 19th
Peczarski, Marcin Piotr, 192. % 11th
Pi ($\pi$), as ``random'' example, 17, 370, 385, 547, 552, 733. % 21st
Preferential arrangements, \see Weak orderings. % 24th
Prodinger, Helmut, 576, 634, 644, 648, 726. %5th
Pyke, Ronald, 732. %4th
Randow, Rabe-R\"udiger von, 606. % 19th
Recursive methods, 114, 214, 218, 243, 313, 350, 452, 592, 596, 713, 715, 717. % 17th
R\'egnier, Mireille, 565, 632. % 19th
Roberts, Charles Sheldon, 573. % 7th
Rothe, Heinrich August, 14, 48, 62, 592. % 10th
Rustin, Randall Dennis, 315, 353. % 21st
Rytter, Wojciech, 454. % 21st
Sagot, Marie-France, 615. % 19th
Samadi, Behrokh ({\arab\Aam/\Afd/\Amm/\Aiss/ \Akh/\Afr/\Amh/\Aib/}), 721. % 7th
Samet, Hanan ({\heb\Htt/\Hmm/\Hss/ \Hfn/\Hnn/\Hkh/}), 566. % 22nd
Sapozhenko, Alexander Antonovich ({\rus Sapozhenko, Aleksandr Antonovich}). % 21st
Schensted, Craige Eugene (= Ea Ea), 57--58, 66. % 7th
Schwartz, Jules Isaac, 128. % 14th
Schwiebert, Loren James, II, 229. % 9th
Seidel, Philipp Ludwig von, 611. % 23rd
Shapiro, Louis Welles, 607. % 19th
{\sl Sefer Yetzirah\/} ({\heb\Hhh/\Hrr/\Hyy/\Hts/\Hyy/ \Hrr/\Hpp/\Hss/}), 23. % 20th
Signed magnitude notation, 177. %4th
Simon, Istv\'an Guszt\'av, 642. % 20th
Singh, Parmanand ({\dn prmAn\kern-0.6em\raise0.4em\hbox{\char21}d Es\kern-0.6em\raise0.4em\hbox{\char21}h}), 270. % 19th
Smith, Wayne Earl, 405. % 19th
Snyder Holberton, Frances Elizabeth, 324, 386, 387. % 12th
Spearman, Charles Edward, 597. % 12th
Sperner, Emanuel, theorem, 744. % 13th
Stable sorting, 4--5, 17, 24, 25, 36--37, 79, 102, 134, 155, 167, 347, 390, 584, 615, 653. % 21st
Stanley, Richard Peter, 69, 600, 605, 606, 670, 671. % 16th
Stasevich, Grigory Vladimirovich ({\rus Stasevich, Grigorii0 Vladimirovich}), 91. % 7th
Successful searches, 392, 396, 532, 550. % 10th
Sue, Jeffrey Yen (\JC48:48:0:40% J7311 edited by hand!
<%
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), 693. % 7th
Symvonis, Antonios ({\grk Sumb'wnhs, Ant'wnios}), 702. % 24th
Tannier, Eric, 615. % 19th
Tokuda, Naoyuki (\JC47:48:0:40% J3833
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