% CHANGES TO FASCICLE V4F3 OF THE ART OF COMPUTER PROGRAMMING
%
% Copyright (C) 2005,2006,2007,2008,2009,2010 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 err4f3"

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

\def\vertical{|}
\def\inref#1 #{\expandafter\def\csname\vertical#1\endcsname}
\catcode`|=\active
\let|\inref \input \jobname.ref
\catcode`|=12

\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
  \line{\kern-5pt\llap{\manfnt x}% print a black triangle in left margin
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  \line{\kern-5pt{\bf Page #2}\enspace #3
   \leaders\hrule\hfill\ \eightrm\date#4.}
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  \medbreak\ninepoint
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   \leaders\hrule\hfill\ \eightrm\date#4.}
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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}
\def\EX #1. [#2]{\ninepoint
  \textindent{\llap{\manfnt x}\bf#1.}[{\it#2\/}]\kern6pt}

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\let\defaultpointsize=\tenpoint

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

\def\lhead{INTRODUCTION}
\let\rhead=\lhead
\titlepage
\volheadline{THE ART OF COMPUTER PROGRAMMING}
\bigskip
\volheadline{ERRATA TO VOLUME 4 FASCICLE 3}
\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~4,
Fascicle~3, since it was first printed in 2005.

\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
\beginconstruction
The ultimate, glorious, future editions of Volumes 1--5 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, June 2005}

\bigskip
\bigskip
{\quoteformat
For a work of such scope as this,
the first word of the author should be an apology
for what is doubtless the too ambitious effort
of a single writer.
\author EDWARD W. BYRN (1900)
% The Progress of Invention in the Nineteenth Century, page i
\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 4 FASCICLE 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\lhead{\kern-12ptCHANGES TO V4F3: GENERATING ALL COMBINATIONS AND PARTITIONS}
\def\rhead{CHANGES TO V4F3: GENERATING ALL COMBINATIONS AND PARTITIONS\kern-12pt}
\titlepage
\volheadline{GENERATING ALL COMBINATIONS AND PARTITIONS} % the fascicle title
\bigskip
\rightline{Copyright \copyright\ 2005, 2006, 2007, 2008, 2009, 2010, Addison\with Wesley; all rights reserved}
\rightline{Last updated \today}
\bigskip
\rightline{\sl Most of these corrections have already been made in
                  recent printings.}
\medskip
\noindent Important note: The changes below include nearly every difference
between the paperback fascicle and the first printing of Volume 4A,
published in January 2011. All subsequent changes can be found in
the errata list for that volume.

\smallskip
\let\defaultpointsize=\tenpoint

\bugonpage 4f3.ii in the ISBN numbers (08.04.29)
line 19: {\tt 1-201-85394-9} \becomes {\tt 0-201-85394-9}\nl
line $-4$:    1-201-85394-9  \becomes      0-201-85394-9
\endchange

\improvepage 4f3.iv line 20 (08.10.28)
I shall happily pay a finder's fee of \$2.56 \becomes
I happily offer a ``finder's fee'' of \$2.56
\endchange

\improvepage 4f3.iv lines 24 and 25 (08.10.28)
reward you with immortal glory instead of mere money \becomes
do my best to give you immortal glory
\endchange

\improvepage 4f3.iv line $-12$ (05.12.04)
indices \becomes indexes
\endchange

\amendpage 4f3.0 replacement for lines 3 and 4 (09.01.12)
\beginconstruction The opening sections of this chapter appear in
Volume~4, Fascicle~0, and
Volume~4, Fascicle~1, first published in 2008 and 2009.\par\endgroup
\endchange

\improvepage 4f3.1 line $-8$ (08.04.21)
The latter representation \becomes The string representation
\endchange

\bugonpage 4f3.2 in the running headline {(also on pages 4, 6, 8,
   \dots!)} (05.09.07)
\eightpoint (F2) \becomes (F3)
\endchange

\amendpage 4f3.2 line 7 (08.10.12)
{\it multicombination}, \becomes
{\it multicombination\/} of $s+1$ things,
\endchange

\amendpage 4f3.4 line 10 (06.10.18)
7.1--00 \becomes 7.1.3--20
\endchange

\improvepage 4f3.4 line $-7$ (05.09.17)
repeat until $c_j+1\ne c_{j+1}$. \becomes
eventually the condition $c_j+1\ne c_{j+1}$ will occur.
\endchange

\improvepage 4f3.5 second line of step T4 (08.03.13)
repeat this step until $x\ne c_{j+1}$ \becomes repeat step T4
\endchange

\bugonpage 4f3.7 in {\eq(23)} (06.07.31)
$w_1\ge w_0;$ \becomes $w_1\ge w_0\ge0@;$
\endchange

\amendpage 4f3.8 lines $-13$ and $-12$ (06.03.24)
discovered by D.~T. Tang \becomes
discovered by J.~E. Miller in her Ph.D. thesis (Columbia University, 1971),
then independently by D.~T. Tang
\endchange

\improvepage 4f3.8 near the bottom (06.10.23)
lines $-3$ and $-1$: increasing \becomes nondecreasing\nl
line $-2$: {\it decreasing} \becomes non{\it increasing}
\endchange

\improvepage 4f3.13 in step C3 (09.08.10)
If $w_j=0$, set $w_j\gets1$, $j\gets j+1$, and repeat until $w_j=1$. \becomes
While $w_j=0$, set $w_j\gets1$ and $j\gets j+1$.
\endchange

\improvepage 4f3.13 in steps C4 and C6 (08.04.21)
$r=j>1$ \becomes $r=j$ and $j>1$
\endchange

\bugonpage 4f3.17 {(and throughout the fascicle!)} (08.04.04)
\ninepoint{\bf Fig.\ 26} \becomes {\bf Fig.\ 46} [and all figure
numbers need to increased by 20, except that there are {\it two\/}
instances of Fig.~35! There are separate errata entries for the
Fig.~35 that becomes Fig.~56.]
\endchange

\amendpage 4f3.19 line $-2$ (10.06.07)
we have \becomes with $\kappa_t@0=0$, we have
\endchange

\improvepage 4f3.20 line $-2$ (05.08.07)
in cross order \becomes in increasing cross order
\endchange

\improvepage 4f3.22 line $-5$ (10.05.30)
In other words, \becomes {\it Proof.}\enspace In other words,
\endchange

\improvepage 4f3.27 exercise 20 (10.05.30)
\ninepoint Find \becomes Devise
\endchange

\amendpage 4f3.27 exercise 21 (06.03.24)
\ninepoint Prove \becomes (Joan E. Miller, 1971.) Prove
\endchange

\bugonpage 4f3.35 in the running headline (05.10.25)
{\eightrm PARTITIONS \becomes COMBINATIONS}
\endchange

\bugonpage 4f3.35 line 3 of exercise 34 (06.03.07)
$11001000,\,11001001$ \becomes $11001000,\,11001010,\,11001001$
\endchange

\amendpage 4f3.35 new exercise (06.08.09)
\ninepoint{\advance\parindent by4pt
\EX111. [M26] (P. Erd\H{o}s, C. Ko, and R. Rado.)
Suppose $A$ is a set of $r$-combinations of an $n$-set, with $\alpha\cap\beta
\ne\emptyset$ whenever $\alpha,\beta\in A$. Show that $\vert A\vert\le
{n-1\choose r-1}$, if $r\le n/2$. \em Hint: Consider $\del^{n-2r}B$, where
$B$ is the set of complements of~$A$.
\par}
\endchange

\amendpage 4f3.36 line 4 (07.08.21)
(see Table 1). All twelve of Stanley's \becomes \nl
(see Table 1), based on a series of lectures by
Gian-Carlo Rota. All twelve of these
\endchange

\amendpage 4f3.37 line 12 (08.09.22)
into disjoint subsets \becomes into nonempty, disjoint subsets
\endchange

\amendpage 4f3.37 bottom line (05.10.24)
52]: \becomes
52], and pad the array with 1s
as suggested by A.~Zoghbi and I.~Stojmenovi\'c [{\sl International
Journal of Computer Math.\ \bf70} (1998), 319--332]:
\endchange

\amendpage 4f3.38 line 3 (06.05.29)
$n\ge1$. \becomes
$n\ge1$. The value of $a_0$ is also set to zero.
\endchange

\amendpage 4f3.38 replacement for step P1 (05.10.24)
\algindentset{P1}%
\algstep P1. [Initialize.] Set $a_m\gets1$ for $n\ge m>1$. Then set $m\gets1$
and $a_0\gets0$.
\endchange

\amendpage 4f3.38 replacement for step P4 (05.10.24)
\algindentset{P1}%
\algstep P4. [Change 2 to $1{+}1$.] Set $a_q\gets1$, $q\gets q-1$, $m\gets
m+1$, and return to P3. (At this point we have $a_k=1$ for $q<k\le n$.)
\endchange

\amendpage 4f3.38 line 3 of Algorithm H (10.05.30)
$n\ge m\ge 2.$ \becomes
$n\ge m\ge 2.$ A sentinel value is stored in $a_{m+1}$.
\endchange

\improvepage 4f3.38 better way to state step H4 (09.08.04)
\algindentset{H1}%
\algstep H4. [Find $j$.] Set $j\gets3$ and $s\gets a_1+a_2-1$. Then, while
$a_j\ge a_1-1$, set $s\gets s+a_j$ and $j\gets j+1$.
(Now $s=a_1+\cdots+a_{j-1}-1$ and $a_j<a_1-1$.)
\endchange

\improvepage 4f3.39 line $-16$ (08.09.22)
$n$-multicombinations of $N$ \becomes $n$-multicombinations of $N$ things
\endchange

\improvepage 4f3.40 line 5 before {\eq(11)} (10.06.14)
by transposing the rows and columns of the corresponding Ferrers diagram.
\becomes
by transposing its
Ferrers diagram\dash---that is, by reflecting the diagram about the
main diagonal.
\endchange

\amendpage 4f3.40 line 6 after {\eq(11)} (08.09.22)
$(a_1\ldots a_m)\losup T$ \becomes
$(a_1\ldots a_m)\losup T$, if $m<n$
\endchange

\amendpage 4f3.40 line 3 before {\eq(13)} (07.08.25)
{\it representation}. \becomes
{\it representation} [see S. Com\'et, {\sl Numer.\ Math.\ \bf1} (1959),
90--109].
\endchange

\amendpage 4f3.41 line 1 (08.09.22)
a path \becomes a path of length~$2n$
\endchange

\improvepage 4f3.41 lines 10 and 11 (10.06.14)
When the partition has $t$ different parts, \becomes
When the number of distinct parts of a partition is equal to~$t$,
\endchange

\amendpage 4f3.41 line 18 (10.08.23)
Philipp \becomes Philippe
\endchange

\improvepage 4f3.42 line 1 (10.09.26)
identity of Jacobi \becomes identity that Jacobi published in 1829
\endchange

\amendpage 4f3.42 the line after {\eq(21)} (10.06.14)
setting $z=e^{-t}$ \becomes setting $z=e^{-t}$ with $t>0$
\endchange

\amendpage 4f3.42 in {\eq(23)} (10.06.14)
$\displaystyle {p(n)\over 1-e^{-t}}$ \becomes
$\displaystyle {p(n)\over 1-e^{-t}}=\sum_{k=n}^\infty p(n)@e^{(n-k)t}$
\endchange

\amendpage 4f3.43 lines 10 and 11 (10.06.14)
$0\le\Re\theta\le1$ \dots~rectangle; \becomes
\endchange

\bugonpage 4f3.43 line 15 (10.06.11)
Theorem 10.6.2 \becomes Theorem 10.6e
\endchange

\amendpage 4f3.43 replacement for lines $-11$ through $-2$ (10.09.27)
$$f(y)\;=\;e^{-{3\over2}t(y+{1\over6})^2+\pi iy};\eqno(27)$$
and this function $f$ qualifies for Poisson's summation formula under both of
the criteria (i) and (ii) stated above. Therefore we try to integrate
$e^{-2\pi miy}f(y)$, and that integral
turns out to be easy for all~$m$ (see exercise 27):
$$\int_{-\infty}^{@\infty} e^{-a(y+b)^2+2ciy}\,dy\;=\;
 \sqrt{\pi\over a}@e^{-c^2\!/a-2bci}\qquad\hbox{when $a>0$.}
\eqno(28)$$
Plugging in to \eq(25), with $\theta=0$, $a={3\over2}t$, $b={1\over6}$,
and $c=({1\over2}-m)\pi$, yields
$$\sum_{n=-\infty}^\infty f(n)=\sum_{m=-\infty}^\infty g(m),\qquad
g(m)=\sqrt{2\pi\over3t}e^{-2(m-\half)^2\pi^2\!/(3t)+{1-2m\over6}\pi i}.
\eqno(29)$$
These terms combine and cancel beautifully, as shown in exercise~27, giving
$${e^{-t/24}\over P(e^{-t})}\;=\;
\sqrt{2\pi\over t}\sum_{n=-\infty}^\infty (-1)^n e^{-6\pi^2(n+{1\over6})^2\!/t}
\;=\;\sqrt{2\pi\over t}{e^{-\zeta(2)/t}\over P(e^{-4\pi^2\!/t})}.\eqno(30)$$
\endchange

\improvepage 4f3.45 line 3 before {\eq(40)} (06.04.18)
by considering \becomes by transposing
\endchange

\improvepage 4f3.46 line 2 below Table 2 (08.03.24)
Erd\H{o}s \becomes Erd\"os \qquad[to agree with the English spelling
in the paper cited]
\endchange

\bugonpage 4f3.47 line 2 before {\eq(47)} (08.09.22)
$\alpha^m=n^{-1/2}e^{-Cx}$ \becomes $\alpha^m=n^{-1/2}e^{-Cx}+O(n\_)$
\endchange

\bugonpage 4f3.48 line 2 after the top illustration (08.09.22)
$O(\sqrt n\,)$ \becomes $O(\sqrt n\log\log\log n)$
\endchange

\amendpage 4f3.48 the line after {\eq(50)} (10.06.14)
$m$ is reasonably large \becomes $m$ and $n$ are reasonably large
\endchange

\amendpage 4f3.49 two lines before {\eq(52)} (10.06.14)
in steps P2 and P6; \becomes
in step P2  or when we'll soon test $n\le x$ in~P6;
\endchange

\bugonpage 4f3.49 line $-1$ (09.05.14)
$4-3C/\sqrt n$ \becomes $3+C/\sqrt n$
\endchange

\amendpage 4f3.50 line 10 (08.09.22)
$(a_1\ldots a_m)^T$ \becomes
$(a_1\ldots a_m)^T$ and $m<n$
\endchange

\improvepage 4f3.51 line 4 after Table~3 (10.05.30)
$a_1a_2\ldots a_n$ \becomes $a_1a_2\ldots{}$
\endchange
\bugonpage 4f3.53 bottom line (06.12.08)
\ninepoint $l_2$, $l_4$, $l_5$, $l_7$ \becomes
$l_2$, $l_4$, $l_5$, $l_6$, $l_7$
\endchange

\amendpage 4f3.54 new wording for exercise 8 (10.06.14)
\ninepoint
\ex 8. [15] When $(p_1\ldots p_t,q_1\ldots q_t)$ yields
the rim representation
of a partition $a_1a_2\ldots{}$ as in \eq(15) and \eq(16), what's the
rim representation of the conjugate partition $(a_1a_2\ldots{})\losup T@$?
\endchange

\bugonpage 4f3.54 new wording for exercise 13 (10.09.26)
\ninepoint
\EX 13. [M23] (F. Franklin, 1882.)
Find a one-to-one correspondence
$\alpha\swap\beta$ between
partitions of~$n$ such that $\alpha$ has exactly $k$ parts
repeated more than once if and only if $\beta$ has
exactly $k$ even parts. (For example, the partition
64421111 has two repeated parts $\{4,1\}$ and three even parts $\{6,4,2\}$.
The case $k=0$ corresponds to Euler's result.)
\endchange

\amendpage 4f3.54 new rating for exercise 14 (10.06.14)
\ninepoint[{\it M28\/}] \becomes [{\it M33\/}]
\endchange

\improvepage 4f3.54 new wording for exercise 16 (08.09.22)
\ninepoint Find \dots\ and sum \becomes
Find a formula for $\sum_{m,n}p(k,m,n)w^mz^n$, where $p(k,m,n)$
is the number of partitions of~$n$ that have $m$~parts and trace~$k$. Sum
\endchange

\amendpage 4f3.55 new sentence for end of exercise {17(b)} (10.06.14)
\ninepoint For example, the
answer when $t=2$ is $(1+v)(1+vz)z^2\!/\bigl((1-z)(1-z^2)\bigr)$.
\endchange

\amendpage 4f3.55 replacement for the last line of exercise 18 (10.06.14)
\ninepoint
$$\sum_{{\scriptstyle a_1\ge\cdots\ge a_r>0,\,r\ge0}_{\mathstrut}^{\mathstrut}
  \atop{\scriptstyle b_1>\cdots>b_s>0,\,s\ge0}} \mkern-45mu
 u^{r+s} v^s z^{a_1+\cdots+a_r+
           b_1+\cdots+b_s}\;=\mkern-15mu
\sum_{{\scriptstyle c_1\ge\cdots\ge c_t>0,\,t\ge0}_{\mathstrut}^{\mathstrut}
  \atop{\scriptstyle d_1,\ldots,d_t\in\{0,1\}}}\mkern-45mu
  u^t v^{d_1+\cdots+d_t}z^{c_1+\cdots+c_t+(t-1)d_1+\cdots+d_{t-1}}.$$
\endchange

\amendpage 4f3.54 new rating for exercise 19 (10.06.14)
\ninepoint[{\it M21\/}] \becomes [{\it M22\/}]
\endchange

\amendpage 4f3.55 line 1 of exercise 21 (08.09.22)
\ninepoint partitions into \becomes partitions of $n$ into
\endchange

\amendpage 4f3.56 in exercise 27 (10.09.27)
\ninepoint [{\it\HM23\/}] Evaluate \eq(29) \becomes
           [{\it\HM21\/}] Prove \eq(28)
\endchange

\amendpage 4f3.57 new wording for exercise 38 (10.06.14)
\ninepoint What is \dots\ largest part $l$? \becomes
Given positive integers $l$ and~$m$, what generating function enumerates
partitions that have exactly $m$~parts, and largest part~$l$?
(See Eq.~\eq(51).)
\endchange

\amendpage 4f3.57 new rating for exercise 40 (10.09.28)
\ninepoint[{\it M22\/}] \becomes [{\it M20\/}]
\endchange

\improvepage 4f3.57 line 1 of exercise 40 (10.09.28)
\ninepoint What is the generating function for partitions \becomes
Continuing exercise 38, what is the generating function
for the number of partitions\nl
[Note: Exercises 39 and 40 will become exercises 40 and 39 in Volume 4A.]
\endchange

\amendpage 4f3.57 new rating for exercise 43 (10.06.18)
\ninepoint[{\it M21\/}] \becomes [{\it M18\/}]
\endchange

\amendpage 4f3.57 in exercise 47 (07.02.23)
\ninepoint line 1 of step N4: 1, 2, \dots, $n$ \becomes 1, 2, \dots,\nl
line $-1$ (the displayed equation): $\displaystyle \sum_{j=1}^m$
\becomes $\displaystyle \sum_{j=1}^\infty$
\endchange

\amendpage 4f3.58 line 3 of exercise 50 (08.09.22)
\ninepoint for all $m$ \becomes for all $m\ge0$.
\endchange

\amendpage 4f3.58 line 1 of exercise 54 (10.06.14)
\ninepoint The partition \dots $b_1b_2\ldots{}$, \becomes
 Let $\alpha=a_1a_2\ldots{}$ and
$\beta=b_1b_2\ldots{}$ be partitions of~$n$. We say that
$\alpha$ {\it majorizes\/}~$\beta$,
\endchange

\bugonpage 4f3.58 line 9 of exercise 55 (10.06.11)
\ninepoint $n_2>n_1\ge0$ \becomes $n_2>n_1\ge0$ and $n_2>2$
\endchange

\amendpage 4f3.59 the rating of exercise 56 (07.01.21)
\ninepoint [{\it M27\/}] \becomes [{\it M32\/}]
\endchange

\amendpage 4f3.60 line 2 of exercise 57 (10.06.14)
\ninepoint Then \becomes By permuting rows and columns we can assume that
$r_1\ge r_2\ge\cdots{}$ and $c_1\ge c_2\ge\cdots{}$. Then
\endchange

\bugonpage 4f3.60 line 2 of exercise 59 (06.10.26)
\ninepoint same as conjugate \becomes same as the conjugate
\endchange

\amendpage 4f3.61 new exercise (09.09.11)
\ninepoint
\ex72. [M30] How many partitions of $n$ have no predecessor in Bulgarian
solitaire?\nl
\smallskip[The former exercise 72 now becomes number 73.]
\endchange

\improvepage 4f3.62 line 18 (05.10.30)
in which we have \becomes of nonnegative integers in which we have
\endchange

\amendpage 4f3.63 an explanatory sentence to follow \eq(9) (08.02.09)
(In other words, we begin with either $A\losub{(m-1)n}(m{-}1)$ or
$A'_{(m-1)n}(m{-}1)$ and then use either
$A_{mn}^R@j$ or $A\losub{mn}@j$, alternately, as $j$ decreases from
$m-1$ to~0.)
\endchange

\amendpage 4f3.64 line $-9$ (10.10.09)
{\sl Annals of Math.}\ \becomes
{\sl Annals of Math.\ \rm(2)}
\endchange

\amendpage 4f3.67 line 2 (08.11.30)
$\max(\Re z,\Im z)$ becomes
$\max\bigl(\vert\Re z\vert,\vert\Im z\vert\bigr)$
\endchange

\bugonpage 4f3.67 lines 6 and 7 (10.11.08)
and if the sums $\sum_{t\in B\setminus A}f(t)$ and
$\sum_{t\in C\setminus A}\hat f(t)$ are both small, then
$\sum_{t\in C}\hat f(t)$ is a good approximation to $\sum_{t\in B}f(t)$.
\becomes\nl
and if the sums $\sum_{t\in B}\vert f(t)\vert$ and
$\sum_{t\in C}\vert\hat f(t)\vert$ are both small, then
$\sum_{t\in A\cup C}\hat f(t)$ is a good approximation to
$\sum_{t\in A\cup B}f(t)$.
\endchange

\bugonpage 4f3.67 line 14 (05.08.23)
$O\Bigl({e^{n-n^{-2\epsilon}\!/2}\over n^n}\Bigr)$ \becomes
$O\Bigl({e^{n-n^{2\epsilon}\!/2}\over n^n}\Bigr)$
\endchange

\amendpage 4f3.67 line $-3$ (08.11.30)
$O(n^{9\epsilon-3/2})$ \becomes $O(n^{9\epsilon-2})$
\endchange

\improvepage 4f3.68 on the line following {\eq(23)} (05.09.17)
saddle point now occurs \becomes saddle point for the new integrand occurs
\endchange

\improvepage 4f3.69 line $-2$ (10.06.17)
$b_0$, $b_1$, $b_2$, $b_3$ \becomes 1, $b_1/n$, $b_2/n^2$, $b_3/n^3$
\endchange

\bugonpage 4f3.72 line $-20$ (09.05.14)
By comparing \eq(47) \becomes
By comparing \eq(46)
\endchange

\improvepage 4f3.74 line 18 (09.05.14)
each $k$-block partition \becomes
each set partition that has $k$ blocks
\endchange

\amendpage 4f3.74 line $-8$ (10.10.18)
different multisets \becomes types of multisets
\endchange

\improvepage 4f3.75 line 22 (08.02.08)
simple \becomes fairly simple
\endchange

\bugonpage 4f3.75 replacement for {\eq(57)} (07.01.18)
$$\openup1\jot
\def\\#1#2{\llap{\smash{\special{ps: .5 setgray}\vrule height 95pt
           \special{ps: 0 setgray}\kern8pt}}%
       \setbox0=\hbox{\ninepoint$f_{#1}=#2$}\hbox to.5em{\hss\rotl0\hss}}
\vcenter{\halign{&\hskip1.5em\hfil$#$\cr
j  &0&1&2&3&4&5&6&7&8&9&\hidewidth10\hidewidth&\hidewidth11\hidewidth\cr
c_j&1&2&1&2&1&2&1&2&1&2&2&2\cr
u_j&9&9&6&8&4&6&2&6&1&5&4&1\cr
v_j&3&1&2&2&2&0&1&1&1&1&3&1\cr
&\\00&&\\12&&\\24&&\\36&&\\48&&\\5{10\ }&\\6{11}&\\7{12}\cr}}\eqno(57)$$
\endchange

\bugonpage 4f3.75 line $-2$ (07.01.18)
$f_0f_1\ldots f_n$, $c_0c_1\ldots c_{n}$,
$u_0u_1\ldots u_{n}$, and $v_0v_1\ldots v_{n}$ \becomes\nl
$f_0f_1\ldots f_n$, $c_0c_1\ldots c_{mn}$,
$u_0u_1\ldots u_{mn}$, and $v_0v_1\ldots v_{mn}$
\endchange

\amendpage 4f3.76 restatement of step M2 (08.02.09)
Set $j\gets a$ and \dots\ until $j=b$. \becomes
Set $j\gets a$,
$k\gets b$, and $x\gets0$. Then while $j<b$ do the following:
Set $u_k\gets u_j-v_j$.
If $u_k=0$, just set $x\gets1$ and $j\gets j+1$.
Otherwise if $x=0$, set $c_k\gets c_j$, $v_k\gets\min(v_j,u_k)$,
$x\gets\[u_k<v_j]$, $k\gets k+1$, $j\gets j+1$.
Otherwise set $c_k\gets c_j$, $v_k\gets u_k$, $k\gets
k+1$, $j\gets j+1$. (Notice that $x=\[\hbox{$v$ has changed}]$.)
\endchange

\amendpage 4f3.76 line 25 (07.01.18)
every component. \becomes every component. (See exercise 68.)
\endchange

\bugonpage 4f3.80 line $-3$ (08.06.06)
\ninepoint Fig.\ 35. \becomes Fig.\ 56
\endchange

\bugonpage 4f3.81 caption to the illustration (08.06.06)
\ninepoint\bf Fig.\ 35. \becomes Fig.\ 56
\endchange

\bugonpage 4f3.81 line 3 after the illustration (08.06.06)
\ninepoint Fig.\ 35. \becomes Fig.\ 56
\endchange

\improvepage 4f3.81 line 7 after the illustration (10.06.17)
Ferrers diagram \becomes tableau shape
\endchange

\bugonpage 4f3.81 in exercise 29 (10.06.06)
\ninepoint line 8: $a_1\ge\cdots\ge a_m$ \becomes
                   $a_1\ge\cdots\ge a_m\ge0$\nl
line 9 (the displayed formula): $y^{a_j+m-j+t}$ \becomes $y^{a_j+j-m+t}$
\endchange

\bugonpage 4f3.82 bottom line of exercise 29 (10.05.14)
\ninepoint
multisets $\{a\losub1{+}m,a\losub2{+}m{-}1,\ldots,
a\losub m{+}1\}$ and $\{a'_1{+}m,a'_2{+}m{-}1,\ldots,a'_m{+}1\}$ \becomes
multisets $\{a\losub1{+}1,a\losub2{+}2,\ldots,
a\losub m{+}m\}$ and $\{a'_1{+}1,a'_2{+}2,\ldots,a'_m{+}m\}$
\endchange

\bugonpage 4f3.85 line 3 of exercise 63 (06.07.01)
\ninepoint $(1@{+}@p_n(1{-}z)$ \becomes $(1@{+}@p_n(z{-}1)$
\endchange

\amendpage 4f3.85 line 1 of exercise 68 (07.01.18)
\ninepoint
[{\it 20\/}] How large can the stack get in Algorithm M, when it \becomes
[{\it 21\/}] How large can variables $l$ and $b$ get in Algorithm M,
when that algorithm
\endchange

\amendpage 4f3.85 line 1 of exercise 69 (08.02.09)
\ninepoint [{\it 21\/}] \becomes [{\it 22\/}]
\endchange

\bugonpage 4f3.86 replacement for dominoes in exercise 82 (05.09.17)
\font\dom=domino scaled 700
$$\chardef\2="B2 \chardef\3="B3 \chardef\6="B6
\def\\#1#2{\vcenter{\offinterlineskip\hbox{\dom#1}\hbox{\dom#2}}}
\\1\2+\\1\3+\\41+\\15+\\1\6\;=\;
\\\2\3+\\4\2+\\\25+\\4\3+\\45\;=\;
\\\2\6+\\5\3+\\\3\6+\\\64+\\\65$$
\endchange

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

\amendpage 4f3.87 replacement for second line of answer 2 (10.03.10)
\noindent result $@$is \ \lower15pt\hbox{\epsfbox{\figdir/v4a.2020}}$\,$.
\vtop{\hsize=92mm\noindent Conversely, we can easily ``read off'' the
representations $a_i$, $b_i$, $c_i$, $d_i$, $p_i$, or $q_i$ of \eq(2)--\eq(11)
from any given path from $(0,0)$ to $(s,t)$.\parfillskip=0pt}
\endchange

\improvepage 4f3.87 in answer 3 (10.01.30)
[change the step numbers F1, F2, F3, F4, F5 respectively to N1, N2, N3, N4, N5]
\endchange

\bugonpage 4f3.87 last line of answer 5 (06.07.31)
$(\delta_t,\ldots,\delta_1)$ \becomes $(\delta_t,\ldots,\delta_0)$
\endchange

\bugonpage 4f3.87 last line of answer 6 (10.05.30)
${n-2\choose t-1}+{n-2\choose t-2}+\cdots+{n-t-1\choose0}$ \becomes
${n-2\choose t-1}+{n-3\choose t-2}+\cdots+{n-t-1\choose0}$
\endchange

\improvepage 4f3.87 step S4 in answer 7 (09.08.10)
set $b_{j-1}\gets b_j-1$, $j\gets j-1$, and repeat until $j=1$. \becomes
set $b_{j-1}\gets b_j-1$ and $j\gets j-1$.
\endchange

\bugonpage 4f3.89 line 5 of answer {12(b)} (10.05.30)
{\bf 10} \becomes {\bf A10}
\endchange

\improvepage 4f3.89 step V4 in answer 9 (09.08.10)
If $a_{k(q+r)}=1$, set
$a_{k(q+r)}\gets0$, $k\gets k+1$, and repeat until $a_{k(q+r)}=0$. \becomes
\endchange

\bugonpage 4f3.90 replacement for answer 19  (06.03.18)
\ans 19. It is the binary representation of post(1000000), where
post$(2^{k+1}-1)=2^k$, and
post$(n)=2^k+{\rm post}(n-2^k+1)$ if $2^k\le n<2^{k+1}-1$,
for $k\ge0$. So it is 11110100001001000100.\tighten
\endchange

\amendpage 4f3.90 addendum to answer 19 (06.10.02)
\noindent [Incidentally, the left-child/right-sibling representation of
$T_\infty$ is the sideways heap.]
\endchange

\amendpage 4f3.90 line 1 of answer 21 (08.10.11)
[See \becomes [Page 40 of Miller's thesis; see also
\endchange

\bugonpage 4f3.91 line 13 of answer 25 (06.04.26)
$\{b_s,\ldots,b_1,c_t\ldots,c_1\}$ \becomes $\{b_s,\ldots,b_1,c_t,\ldots,c_1\}$
\endchange

\bugonpage 4f3.92 line 6 of answer 29 (10.06.06)
$1\le j\le t$ \becomes $1\le j\le s$
\endchange

\bugonpage 4f3.92 line $-2$ of answer 30 (10.05.30)
$\widehat C_{st}$ is $A_{\lfloor2^{s-\?1}\!/3\rfloor}(s,t)$ \becomes
$\widehat C_{st}$ is $A_{\lfloor2^{s-\?1}\!/3\rfloor}(s,t)^R$
\endchange

\amendpage 4f3.95 addendum to answer 43 (06.12.08)
(Incidentally, $s(x)=f(x)\xor1$ and $p(x)=f(x\xor1)$,
where $f(x)=(x\dotminus1)+((x\band1)\lsh1)$.)
\endchange

\bugonpage 4f3.97 line 1 of answer 55 (05.09.06)
[Solution \becomes (a) [Solution
\endchange

\bugonpage 4f3.97 line 4 of answer 55 (05.12.03)
$t\le i\le n$ \becomes $t\le j\le n$
\endchange

\amendpage 4f3.98 line 5 (10.01.26)
576.] \becomes 576; see~also A.~Williams, {\sl SODA\/ \bf20}
(2009), 987--996, for a significant
generalization by which the permutations of an arbitrary multiset
can be generated looplessly by prefix rotations.]
\endchange

\bugonpage 4f3.98 line 4 of answer 60 (08.02.07)
$x=t$ \becomes $x\gets t$.
\endchange

\improvepage 4f3.98 in step Q2 of answer 60 (09.08.10)
$j\gets j+1$, and repeat until $x\le m_j$. \becomes
and $j\gets j+1$.
\endchange

\bugonpage 4f3.99 line $-6$ of answer 62 (10.05.19)
$r_{i+1}+\cdots+r_m-c_1-\cdots-c_{j-1},c_1+\cdots+c_j-r_i-\cdots-r_m$ \becomes
\nl\quad $r_1+\cdots+r_i-c_{j+1}-\cdots-c_n,c_j+\cdots+c_n-r_1-\cdots-r_{i-1}$
\endchange

\improvepage 4f3.99 line $-5$ of answer 62 (05.09.06)
[This line, the beginning of part (e), should be indented.]
\endchange

\improvepage 4f3.101 line 1 of answer 70 (10.05.30)
From the identity \becomes Using the facts that $t\ge3$ implies
\endchange

\bugonpage 4f3.101 line 3 of answer 70 (10.06.06)
${2t-2\choose t}{1\over t}$ \becomes
${2t-2\choose t}{1\over t-1}$
\endchange

\bugonpage 4f3.102 line $-4$ of answer 77 (10.05.30)
${n_v+1\choose v}$ \becomes ${n_v+1\choose v+1}$
\endchange

\bugonpage 4f3.106 last line of answer 97 (10.06.09)
$13300\swap14400$ \becomes $13300\swap14310$
\endchange

\bugonpage 4f3.108 line 2 of answer 107 (06.08.09)
3-combinations \becomes 2-combinations
\endchange

\amendpage 4f3.109 new answer (06.08.09)
{\advance\parindent by4pt
\ans 111. $\del^{n-2r}B$ is the set of all $r$-subsets of $B$; these subsets
must not be in~$A$. If $\vert A\vert=\break\vert B\vert=
{x\choose n-r}$ for some real $x>n-1$,
we would have ${n\choose r}\ge\vert A\vert+\vert\del^{n-2r}B\vert\ge
{x\choose n-r}+{x\choose r}>{n-1\choose n-r}+{n-1\choose r}={n\choose r}$,
by exercise 80. [See {\sl Quart.\ J. Math.\ Oxford\/ \bf12} (1961),
313--320.]
\par}
\endchange

\bugonpage 4f3.110 in the running headline (05.08.27)
7.2.1.3 \becomes 7.2.1.4
\endchange

\improvepage 4f3.110 line 1 of answer 4 (10.06.11)
We must \becomes Assume that $1\le r\le n$. We must
\endchange

\improvepage 4f3.110 simplification of step C4 in answer 5 (10.06.18)
$c_2\gets c2+1$. Then \dots~$l_1\gets2$. \becomes
$c_2\gets c_2+1$, and $l_{@[c_1>0]}\gets2$. % \[...]
If $k\ne2$, also set $l_2\gets k$. Return to~C2.
\endchange

\amendpage 4f3.110 improved answer 6 (10.09.27)
\ans6. Assume that each partition is followed by 0.
Set $j\gets0$, $k\gets a_1$, and $b_{k+1}\gets0$. Then,
while $k>0$, set $j\gets j+1$ and, while $k>a_{j+1}$, set
$b_k\gets j$ and $k\gets k-1$.
(We have used \eq(11) in the dual form $a_j-a_{j+1}=
d_j$, where $d_1\ldots d_n$ is the part-count representation of
$b_1b_2\ldots{}@$. This algorithm essentially walks along the rim
of the Ferrers diagram; so its running time is roughly
proportional to $a_1+b_1$, the number of parts in the output
plus the number of parts in the input.)
\endchange

\amendpage 4f3.110 last line of answer 8 (10.06.14)
$b_1b_2\ldots{}$ \becomes $(a_1a_2\ldots{})\losup T$
\endchange

\bugonpage 4f3.111 line 3 of answer 11 (08.09.22)
infinite series $\prod_{k\ge1_{\mathstrut}}(1+z\losup k+z\losup{2k})
 (1+z\losup{2k}+z\losup{4k})
 (1+z\losup{5k}+z\losup{10k})$ \becomes \nl
infinite product $\prod_{k\ge0}\prod_{@r\in\{10^k,2\cdot10^k,5\cdot10^k\}}
 (1+z^r+z^{2r})$
\endchange

\amendpage 4f3.111 replacement for answer 13 (10.09.26)
\ans13. Map the partition $c_1{\cdot}1+c_2{\cdot}2+c_3{\cdot}3+\cdots{}$ into
$r_1{\cdot}1+\lfloor c_1/2\rfloor{\cdot}2+
 r_3{\cdot}3+\lfloor c_2/2\rfloor{\cdot}4+
 r_5{\cdot}5+\lfloor c_3/2\rfloor{\cdot}6+\cdots{}$,
where $r_m=(c_m\mod2)+2(c_{2m}\mod2)+4(c_{4m}\mod2)+
8(c_{8m}\mod2)+\cdots{}$;
$433222211\mapsto64421111\mapsto8332211$.
[{\sl Johns Hopkins Univ.\ Circular \bf2}
(1882), 72.]\tighten
\endchange

\bugonpage 4f3.111 line 2 of answer 14 (08.09.22)
odd permutation are centered and divided \becomes
odd parts are centered and the partition is divided
\endchange

\amendpage 4f3.111 replacement for second paragraph of answer 14 (08.06.18)
In general, when the ``Durfee rectangle'' (shown in the diagram) has
$t$~rows, suppose there are $a_1$, \dots,~$a_t$ extra dots at the right
and $b_t$, \dots,~$b_1$, \dots, $b_t$ extra dots below, where
$a_1\ge\cdots\ge a_t\ge0$ and $b_1\ge\cdots\ge b_t\ge0$. Then the distinct
parts obtained are $2t-1+a_1+b_1$, $2t-2+a_1+b_2$, \dots, $2+a_{t-1}+b_t$,
$1+a_t+b_t$, and (if it's nonzero) $0+a_t$. Conversely, any partition
with $2t$ distinct nonnegative parts can uniquely be written in this form.
\endchange

\amendpage 4f3.112 line 10 of answer 17 (06.10.23)
``Partition bijections, a survey,'' to appear in {\sl The Ramanujan Journal}.
\becomes
{\sl The Ramanujan Journal\/ \bf12} (2006), 5--75.
\endchange

\bugonpage 4f3.112 line $-3$ of answer 17 (10.06.18)
$r=0$ \becomes $s=0$
\endchange

\bugonpage 4f3.112 line $-3$ of answer 17 (06.07.31)
$b_1<t$).\ \becomes $b_1<t$.
\endchange

\amendpage 4f3.113 replacement for first paragraph of answer 23 (10.06.18)
\ans23. (Solution by Marc van Leeuwen.) Divide \eq(19) by $1-v$, to get
$$\eqalign{\prod_{k=1}^{@\infty}(1-u^kv^{k-1})@(1-u^kv^k)@(1-u^kv^{k+1})
&\;=\;\sum_{n=0}^\infty(-1)^n@u^{n+1\choose2}\biggl(
 {v^{n\choose 2}-v^{n+2\choose2}\over1-v}\biggr)\cr
&\;=\;\sum_{n=0}^\infty(-1)^n@u^{n+1\choose2}\sum_{k=0}^{2n}v^k;\cr}$$
now set $u=z$ and $v=1$.
\endchange

\bugonpage 4f3.114 line 2 of answer 24 (08.09.22)
summed over all integers $j$ and $k$ \becomes
summed over all integers $j$ and all nonnegative integers~$k$
\endchange

\bugonpage 4f3.114 line 7 of answer 24 (08.09.22)
$/P(z^5)$ \becomes $/P(z^5)^5$
\endchange

\bugonpage 4f3.114 line 4 of answer 26 (08.09.22)
$e^{-\pi^2m^2n^2}$ \becomes $e^{-\pi^2m^2n}$
\endchange

\amendpage 4f3.114 Replacement for answer 27 (10.09.27)
\ans27. First, $\int_{-\infty}^\infty e^{-a(y+b)^2+2ciy}\,dy=
e^{-c^2\!/a-2bci}\int_{-\infty}^\infty e^{-au^2}\,du$, by the substitution
$u=y+b-ci/a$. And $\int_{-\infty}^\infty e^{-au^2}\,du=\int_0^\infty
e^{-t}\,dt/\sqrt{at}=\Gamma(\half)/\sqrt a=\sqrt{\pi/a}$, by the
substitution $t=au^2$ and exercises 1.2.5--20, 1.2.6--43.\par\breathe
Now \eq(30) follows from (29) because we have, for all integers $m$,
$$g(3m+1)+g(-3m)=\sqrt{{2\pi\over t}}(-1)^m e^{-6\pi^2(m+{1\over6})^2\!/t}@;
\quad g(3m+2)+g(-3m-1)=0.$$
[See M.~I. Knopp, {\sl Modular Functions in Analytic Number Theory\/}
(1970), Chapter~3.]
\endchange

\bugonpage 4f3.115 answer 28 (08.09.22)
line $-2$: If $n={}$ \becomes If $k={}$\nl
line $-1$: $2^{-t}\bigl(1+(-1)^{[e_1=1]}/p_1\bigr)\ldots
              \bigl(1+(-1)^{[e_t=1]}/p_t\bigr)$ \becomes\nl\qquad\qquad
           $2^{-t}\bigl(1+(-1)^{[e_1>1]}/p_1\bigr)\ldots
              \bigl(1+(-1)^{[e_t>1]}/p_t\bigr)$
\endchange

\improvepage 4f3.115 line 7 of answer 31 (10.04.18)
$f_{pq}$ \becomes $f_{p,q}$\qquad (twice)
\endchange

\bugonpage 4f3.115 line 7 of answer 31 (08.09.22)
less than $m/q$ \becomes less than $\lfloor m/q\rfloor$
\endchange

\bugonpage 4f3.115 line 2 of answer 33 (08.09.22)
$m=\sqrt n$ \becomes $m=\lfloor\sqrt n\rfloor$
\endchange

\bugonpage 4f3.116 last line of answer 37 (06.08.26)
{\sl de Scienze Economiche e Commerciale\/} \becomes
{\sl di Scienze Economiche e Commerciali\/}
\endchange

\amendpage 4f3.116 new answer 40 (10.09.28)
\ans40. $[x^m]\,(1+zx)(1+z^2x)\ldots(1+z^{l-1}x)=
z^{m(m+1)/2}{l-1\choose m}{\atop\!\?z}$, by exercise 1.2.6--58; this is
$(z-z^l)(z^2-z^l)\ldots(z^m-z^l)/
((1-z)(1-z^2)\ldots(1-z^m))$.
The answer also follows from Theorem~C\null: Replacing $a_1\ldots a_m$ by
$(a_1-m)\ldots(a_m-1)$ gives an equivalent partition of $n-m(m+1)/2$ into at
most $m$ parts, not exceeding $l-1-m$.
\endchange

\amendpage 4f3.117 new answer 43 (10.06.18)
\ans43. $p\bigl(n-k(k-1)/2\bigr)$. (Change $a_1a_2\ldots a_k$ to
$(a_1-k+1)(a_2-k+2)\ldots a_k$ to get an equivalent partition
of $n-k(k-1)/2$.)
\endchange

\improvepage 4f3.118 part {(e)} of answer 50 (10.05.30)
line 1: The \becomes Continuing the previous exercise and its answer, the\nl
lines 2 and 3: $(1-z)$, where $F_k(z)$ is defined in the previous
exercise. \becomes $(1-z)$.
\endchange

\bugonpage 4f3.118 last line of answer 53 (07.02.07)
20~8~8~8~6~5~4~3~3~3~3~2~$1^{19}$ \becomes
20~8~8~8~8~6~5~4~3~3~3~3~2~$1^{19}$
\endchange

\bugonpage 4f3.118 part {(b)} of answer 54 (05.11.05)
$n-kl$ \becomes $n+kl$\qquad\qquad(three changes)
\endchange

\bugonpage 4f3.118 line 3 of answer 54 (10.06.18)
$a'_k$ \becomes $a'_{@l}$
\endchange

\bugonpage 4f3.118 the end of answer 54{(c)} (05.11.05)
$0\le\cdots\le c_{k+1}\le\cdots c_k$ \becomes
$0\le\min(a_{k+1},b_{k+1})\le
     \min(a_{k+1},b_{k+1}+b_1+\cdots+b_k-a_1-\cdots-a_k)
=c_{k+1}\le a_{k+1}\le a_k=c_k+(c_1+\cdots+c_{k-1})
-(a_1+\cdots+a_{k-1})\le c_k$.
\endchange

\bugonpage 4f3.118 in part {(f)} of answer 54 (10.06.18)
False \dots~$(17\,16\,5\,5\,4)$. \becomes
False for $\alpha\vee\beta@$; for example, $6321\vee543=633$ in Fig.~52.
\endchange

\amendpage 4f3.118 and 119, simplification of answer {55(a)} (10.06.11)
Find the smallest $k$ \dots $\[k=l'+1]$. \becomes
Find the smallest~$k$ with $a_k>b_k$,
the smallest $l$ with $a_k>a_{@l+1}$, and the smallest $m$ with
$a_k-1>a_{m+1}$. (Note that $b_k>0$.)
If $a_m>a_{m+1}+1$, define
$\gamma=c_1c_2\ldots{}$ by $c_k=a_k-\[k=m]+\[k=m{+}1]$.
Otherwise let $c_k=a_k-\[k=l]+\[k=m{+}1]$.
\endchange

\bugonpage 4f3.120 line $-3$ of answer 57 (08.09.22)
40--70 \becomes 49--70
\endchange

\bugonpage 4f3.121 line $-1$ of answer 59 (08.09.22)
(see exercise~16) \becomes (see exercise~15)
\endchange

\amendpage 4f3.122 line $-2$ of answer 64 (09.07.23)
to appear.\ \becomes
 {\tt http:/\kern-.1em/arxiv.org/abs/0907.3873\kern.1em}.
\endchange

\improvepage 4f3.122 line $-7$ of answer 66 (08.09.22)
for all such \becomes for the number of all such
\endchange

\bugonpage 4f3.123 line 5 of answer 68 (08.09.22)
No part \becomes Assume that $n>1$. Then no part
\endchange

\amendpage 4f3.124 lines 12--13 (09.09.11)
The origin of the problem is unknown; see \becomes
The problem reportedly stems from Russia via Bulgaria and Sweden.
See also
\endchange

\amendpage 4f3.124 new answer (09.09.11)
\ans72. Equivalently, $a_1<m-1$ [B. Hopkins and J.~A. Sellers,
{\sl Integers\/ \bf7},\thinspace2 (2007), \#A19].\nl
\smallskip[The former answer 72 now becomes number 73.]
\endchange

\amendpage 4f3.124 line 1 of answer 73 {(formerly 72)} (07.08.21)
Swap \becomes [R. Stanley, 1972 (unpublished).] Swap
\endchange

\amendpage 4f3.125 line 4 of answer 7 (07.04.25)
``Partially ordered \dots~to appear \becomes
{\sl Discrete Math.\ \bf298} (2005), 212--229
\endchange

\amendpage 4f3.131 lines 5--7 (07.04.25)
in the paper ``Crossings \dots~partitions'' \becomes in a paper\nl
(preprint, 2005) \becomes
[{\sl Transactions of the Amer.\ Math.\ Soc.\ \bf359} (2007), 1555--1575]
\endchange

\amendpage 4f3.131 line $-2$ of answer 27 (05.11.05)
S. Fomin, \becomes
A.~Berele, {\sl J. Combinatorial Theory\/ \bf A43} (1986), 320--328;
S.~Fomin,
\endchange

\bugonpage 4f3.132 line 3 of answer 29 (08.06.06)
example of Fig.\ 35 \becomes example of Fig.\ 56
\endchange

\bugonpage 4f3.132 line 23 of answer 29 (06.06.27)
[The formula \becomes \em Notes: The formula
\endchange

\bugonpage 4f3.134 last line of answer 35 (08.10.19)
{\bf 58} (1954) \becomes {\bf 58} (1952)
\endchange

\bugonpage 4f3.138 line $-4$ of answer 53 (08.06.06)
See P. L. Chebyshev \becomes [See P. L. Chebyshev
\endchange

\amendpage 4f3.141 lines 2 and 3 of answer 68 (07.01.18)
The maximum level is also equal to~$n$. \becomes
Then the values of $l+1=n$ and $b=mn_1+(m-1)n_2+\cdots+n_m$ 
reach their maximum.
[Thus it's best to choose names of the multiset elements
so that $n_1\le n_2\le\cdots\le n_m$.]
\endchange

\bugonpage 4f3.142 bottom line of answer 76 (10.06.17)
exercise 75 \becomes exercise 73
\endchange

\bugonpage 4f3.143 replacement for domino displays in answer 82 (05.09.17)
\begingroup\chardef\2="B2 \chardef\3="B3 \chardef\6="B6
\font\dom=domino scaled 500
\def\\#1#2{\vcenter{\offinterlineskip\hbox{\dom#1}\hbox{\dom#2}}}%
$$\thickmuskip=8mu
\\1\2\equiv\\\24\equiv\\\3\6\,,\quad
\\1\3\equiv\\\2\6\,,\quad
\\\2\3\equiv\\4\6\,.$$
$$\\\21+\\15+\\4\2+\\5\3+\\\6\3\;=\;
\\1\3+\\\61+\\\25+\\\2\6+\\45\;=\;
\\41+\\\2\3+\\4\3+\\4\6+\\\65$$
\medskip
\endgroup
\endchange

\expandafter\ifx\csname indexeject\endcsname\relax\else\vfill\eject\fi
\amendpage 4f3.144 and following (05.06.23)
Miscellaneous changes to the existing index of Volume~4 Fascicle~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:}
\hangindent2em
Becker, Harold William, 129, 134. % 2nd
Bell numbers, 64--65, 80--84, 123, 125. % 6th
Berele, Allan, 131. % 2nd
Bitstring notation: Data represented as a string of $0@$s and~1s, 1--2, 26, 28. % 2nd
Bitwise operations, 4, 95, 109. % 2nd
Catalan numbers, 101. % 5th
Com\'et, Stig, 40. % 2nd
Dominance order, \see Majorization. % 2nd
Erd\H{o}s, P\'al (= Paul), 19, 35, 46, 57. % 2nd
$f$-vectors, \see Size vectors. % 2nd
Ferrers diagrams, 39--40, 45, 48, 51, 72, 118, 120, 123, 131. % 6th
Fibonacci, Leonardo, of Pisa (= Leonardo filio Bonacii Pisano), recurrence, 42. % 2nd
Fomin, Sergey Vladimirovich ({\rus Fomin, Sergei0 Vladimirovich}), 131. % 2nd
Goldman, Jay Robert, 132. % 2nd
Hickey, Michael, 16, 97. % 2nd
Inclusion and exclusion principle, 46, 57, 134, 139. % 2nd
Ko, Chao (= Ke Zhao, \GC74:74:-3:61% G3134
<001e0000000000000000001fc000000000000000000ff000000000000000000ff0000000%
 00000000000ff000000000000000000ff000000000000000000f8000000000001c00000f%
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 0000001f8000000f81c00000001f8000000f83e00000001f8000000f87f00000001f8000%
 fffffff80000001f80007ffffffc0000001f80003ffffffc0000001f8000080f80038001%
 c01f8000000f8003c003e01f8000000f8003f007f01f8000001f8003fffff81f8000001f%
 8003fffffc1f8000001f8003f003e01f8000001f8001f003e01f8000001f8001f003e01f%
 8000001f8001f003e01f8000003f8001f003f01f8000003ff001f003f01f8000007ffe01%
 f003f01f8000007fff81f003f01f8000007fffc1f003f01f800000ff9fe1f003f01f8000%
 00ff8ff1f003f01f800000ff87f1f003f01f800001ef83f9f003f01f800001ef81f9f003%
 f01f800003cf80f9f003f01f800003cf80f9f003f01f8000078f8079f003f01f8000078f%
 8021f003f01f80000f0f8001f003f01f80000f0f8001f003f01f80001e0f8001fffff01f%
 80001e0f8001fffff01f80003c0f8001f003f01f80007c0f8003f003f01f8000780f8003%
 f003f01f8000e00f8003f003f01f8000c00f8003f003e01f8000800f8003f000001f8000%
 800f8003f000001f8000000f8003f000001f8000000f80000000001f8000000f80000000%
 001f8000000f80000000001f8000000f80000000001f8000000f80000000001f8000000f%
 80000000001f8000000f80000000001f8000000f80000000001f8000000f80000000001f%
 8000000f80000000ffff8000001f80000000ffff8000001f800000001fff8000001f8000%
 000007ff0000001f8000000003ff0000001f8000000001ff0000001f8000000000fe0000%
 001f0000000000fe0000001f0000000000f00000>%% Unicode char "67ef
\GC68:72:-5:59% G5357
<000000000000000e00000000000000003f00000000000000007f801fffffffffffffffc0%
 0fffffffffffffffe007fffffffffffffff00200001fc000003ff00000001fc000003fc0%
 0000001f8000003f000000003f8000003f000000003f8000003f000000003f8000003f00%
 0000003f8000003f000000003f8000007f000000003f0000007f000000003f0000007f00%
 0000007f0000007e000000007e0000007e00000000fe000000fe00000000fc000000fe00%
 000001fc000001fe00000001f8000001fc00000003f8000001fc00000003f0000001f800%
 000007f001f803f80000000fe001ff83f00000000fc0001ffff00000001f800007ffe000%
 00003f800003ffe00000007f000001ffc00000007e000000ff80000000fc000000ff0000%
 0001fc0000007e00000003f80000004000000007f0000000000000000fe000000000e000%
 001ffc00000001f800003e7f00000003fc00007c7ffffffffffe0001f07ffffffffffe00%
 07e07ffffffffffe000f007e00000001f8003c007e00000001f800e0007e00000001f800%
 c0007e00000001f80000007e00000001f80000007e00000001f80000007e00000001f800%
 00007e00000001f80000007e00000001f80000007e00000001f80000007e00000001f800%
 00007e00000001f80000007e00000001f80000007e00000001f80000007e00000001f800%
 00007e00000001f80000007e00000001f80000007e00000001f80000007e00000001f800%
 00007ffffffffff80000007ffffffffff80000007ffffffffff80000007e00000001f800%
 00007e00000001f80000007e00000001f80000007e00000001f80000007e00000001f800%
 00007e00000001f8000000fe00000001f8000000fc00000001f8000000fc00000001f000%
 >%%  Unicode char "53ec
), 35. % 2nd
Leeuwen, Marcus Aurelius Augustinus van, 113, 131. % 6th
Left-child/right-sibling links, 90. % 2nd
Lenin, Vladimir Ilyich ({\rus Lenin, Vladimir~Ilp1ich}), 11. % 2nd
Littlewood, Dudley Ernest, 120. % 2nd
Littlewood, John Edensor, 121. % 2nd
Loizou, Georghios (= George; {\grk Lo"'izou, Ge'wrgios}), 110, 111. % 2nd
Loopless generation, 8, 25, 27, 28, 92, 96--98, 110. % 4th
Miller, Joan Elizabeth, 8, 27. % 2nd
Motzkin, Theodor (= Theodore) Samuel \indexbreak
 ({\heb\Hfn/\Hyy/\Hqq/\Hts/\Hvv/\Hmm/
 \Hll/\Haa/\Hvv/\Hmm/\Hsh/ \Hrr/\Hvv/\Hdd/\Hvv/\Haa/\Hyy/\Htv/}), 77, 128, 141. % 2nd
Multisets, permutations of, 4, 14--15, 29, 30, 41, 89, 98, 123. % 4th
Naud\'e, Philippe, junior, 41. % 6th
Notational conventions:\indexnoperiod % 2nd
\sub $\Gamma^R$ (reversal), 8. % 2nd
\sub ${n\parts m}^{\mathstrut}_{\mathstrut}$ (the number of partitions of~$n$ into exactly $m$~parts), 38. % 2nd
Permutations, of a multiset, 4, 14--15, 29, 30, 41, 89, 98, 123. % 4th
Rad\'o, Richard, 35. % 2nd
Rota, Gian-Carlo, 36. % 2nd
Seth, Vikram ({\dn Evk\kern-1em\char125\kern.3emm s\kern-0.8em\char3W}), ii, 83. % 2nd
Sideways heap, 90. % 2nd
Stojmenovi\'c, Ivan Dan\v{c}a ({\rus Stojmenovits1, Ivan Dancha}), 37. % 2nd
Tableau shapes, 40, 81, \see Ferrers diagrams. % 6th
Ulyanov, Vladimir Ilyich ({\rus Ulp1yanov, Vladimir Ilp1ich}), 11. % 2nd
van Leeuwen, Marcus Aurelius Augustinus, 113, 131. % 6th
Williams, Aaron Michael, 97--98. % 4th
Zoghbi, Antoine Chaiban \indexbreak
 ({\arab\Afy/\Amb/\Aigh/\Afz/\Ail/\Aa/ \An/\Afa/\Amb/\Amy/\Aish/ \An/\Afw/\Amtt/\Ain/\Aha/}), 37. % 2nd

\vfill
\enddoublecolumns
\endchange

\bye

[The next printing would have been the 6th.]
not listed above:
page 38, I've changed the introductory remarks at beginning of algs P and H
page 42, I've qualified (20) to consider the cases of n<=0
page 61, new phrase in line 3 of exercise 73
page 136, line 1 better spacing (H1, ...)

check |gosper-hack|=20 in 7.1.3.ref, regarding page 4f3.4

ARTICLES "TO APPEAR" THAT ARE STILL PENDING: