% ejc assigned name for pdf submission: 2571-3600-1-SM.pdf
\documentclass[12pt]{article}
\usepackage{e-jc}
\usepackage{graphicx}
\usepackage{amsthm,amssymb}
{\newtheorem{definition}{Definition}
\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}}
{\theoremstyle{definition}
\newtheorem{example}{Example}
\newtheorem*{remark}{Remark}
\newtheorem*{conventions}{Conventions}}
\def\BeginExample{\begin{example}}
\def\EndExample{\end{example}}
\def\BeginRemark{\begin{remark}}
\def\EndRemark{\end{remark}}
\def\BeginProof{\noindent{\bf Proof}: }
\def\BeginProofOf#1{\noindent{\bf Proof of #1}: }
\def\EndProof{\QED\betweenskip}
\def\betweenskip{\vskip10pt}
\def\boxitat#1#2{\vbox % from TeXbook Ex.21.3
{\hrule\hbox{\vrule\kern#1%
\vbox{\kern #1\hbox{#2}\kern#1}\kern#1\vrule}\hrule}}
\def\enclose#1{\boxitat{0pt}{#1}}
\def\qedspace{\null}
\def\qedblack{\qedspace % SOLID BOX
\lower.6pt\hbox{\vrule height7pt width 5pt}}
\def\qedwhite{\qedspace % HOLLOW BOX
\lower.6pt\enclose{%
\hbox{\vrule height7pt width 0pt\hskip6pt}}}
\def\QED{\qedwhite}
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\naturals{{\mathbb N}}
\def\integers{{\mathbb Z}}
\def\partsum{\Sigma}
\def\len{{\rm len}}
\def\lrf{\phi} % local restriction function
\def\lrfs{\Phi} % set of lrf's
\def\set#1{{\cal #1}}
\def\mod{{\rm \,mod\,}}
\def\p{u}
\def\pp{v}
%% for compositions we use boldface with arrow
\def\cvec#1{{\vec{\mathbf{#1}}}}
\def\Tr{{\rm t}}
\def\free{\varphi}
%% this might become obsolete
\def\rulefont#1{{\mathfrak{#1}}}
%% cal letters
\def\sC{{\cal C}} % a locally restricted class of compositions
\def\F{{\cal F}} % a family of generating functions
\def\NC{C} % \NC(n) is the number of compositions of n
\def\L{{\mathbf L}} % denotes a particular k-tuple
\def\bfR{{\mathbf R}} % denotes a k-tuple
\def\calM{{\cal M}} % denotes a set of compositions
\def\eps{\epsilon}
\def\|{\parallel}
\setlength{\arraycolsep}{3pt} % smaller array separations (5pt default)
%% probability notation
\def\Ex{{\mathsf E}}
\def\km{{\lfloor \log m-\omega(m) \rfloor}}
\def\kp{{\lfloor \log m+\omega(m) \rfloor}}
\def\kmn{{\lfloor \log n-\omega(n) \rfloor}}
\def\kpn{{\lfloor \log n+\omega(n) \rfloor}}
\title {Locally Restricted Compositions IV.\\
Nearly Free Large Parts and Gap-Freeness}
\author{Edward A. Bender\\
\small Department of Mathematics\\[-0.8ex]
\small University of California, San Diego\\[-0.8ex]
\small La Jolla, CA 92093-0112\\
\small\tt ebender@ucsd.edu\\
\and
E. Rodney Canfield\thanks
{Research supported by NSA Mathematical Sciences Program.}\\
\small Department of Computer Science\\[-0.8ex]
\small University of Georgia\\[-0.8ex]
\small Athens, GA 30602\\
\small\tt erc@cs.uga.edu\\
\and
Zhicheng Gao\thanks
{Research supported by NSERC.}\\
\small School of Mathematics and Statistics\\[-0.8ex]
\small Carleton University\\[-0.8ex]
\small Ottawa, Ontario K1S5B6\\
\small\tt zgao@math.carleton.ca\\}
\date{\dateline{July 19, 2012}{?}{?}\\
\small AMS Subject Classification: 05A15, 05A16}
\begin{document}
\maketitle
\begin{abstract}
We define the notion of asymptotically free for locally restricted
compositions, which means roughly that large parts can often be
replaced by any larger parts.
Two well-known examples are Carlitz and alternating compositions.
We show that large parts have asymptotically geometric distributions.
This leads to asymptotically independent Poisson variables for numbers of
various large parts.
Based on this we obtain asymptotic formulas for the probability of
being gap free and for the expected values of the largest
part, number of distinct parts and number of parts of multiplicity $k$,
all accurate to $o(1)$.
\end{abstract}
\centerline{\it Dedicated to the memory of Herb Wilf.}
\section{Introduction}
Various authors have considered aspects of unrestricted compositions
and Carlitz compositions (unequal adjacent parts) that require knowledge
about the large parts.
The results include information about largest part, number of distinct
parts, gap-freeness and number of parts of multiplicity $k$.
We extend these results to a broad class of compositions, drawing on
earlier work on {\em locally restricted compositions}~\cite{BC2} by
defining a subclass of locally restricted compositions for which
we can show that the large parts are asymptotically independent
geometric random variables.
This leads to asymptotically independent Poisson random variables
for numbers of various large parts.
Our main goal is to prove Theorem~\ref{thm:main}.
Although a full understanding of the theorem requires some definitions,
it can be read now.
Among the compositions included in our definition are unrestricted,
Carlitz and alternating up-down.
Although it was not possible to compute generating functions in~\cite{BC2},
various properties were established, including the following.
\begin{itemize}
\item[(a)]
The number of compositions of $n$ is
$Ar^{-n}(1+O(\delta^n))$ for some $0<\delta<1$ because of a simple
pole in the generating function. Since the convergence to
$Ar^{-n}$ is exponentially fast, the values of $r$ and $A$ can be estimated
fairly easily if one can count compositions for relatively small
values of $n$.
\cite[Theorem~3]{BC2}
\item[(b)]
If a subcomposition can occur
arbitrarily often, the number of times it occurs in a random
composition of $n$ has a distribution that is asymptotically
normal with mean and variance asymptotically proportional to $n$.
The same is true for the total number of parts in a random
composition.
\cite[Theorem~4]{BC2}
\item[(c)]
In many cases, the largest part and number
of distinct parts in a random composition is asymptotic to
$\log_{1/r}n$.
\cite[Section~9]{BC2}
\end{itemize}
Various special cases were considered ~\cite{BC1,BC3}, where more
could be said about the generating functions.
In none of these papers was the behavior of the large parts addressed beyond that
in~(c).
\begin{definition}
[Composition terminology]
\label{definition:composition terminology}
$\naturals$ and $\naturals_0$ denote the positive integers and the non-negative
integers, respectively.
A composition is written $\cvec c=c_1\cdots c_k$ where $c_i\in\naturals$.
We use the same notation to denote concatenation of compositions as
in $\cvec a_1\cdots\cvec a_m$. The length of $\cvec c=c_1\cdots c_k$ is denoted by $\len(\cvec c)=k$
and the sum of the parts by $\partsum(\cvec c)$.
A {\em subcomposition} of $\cvec c$ is a sequence of one or more
consecutive parts of $\cvec c$.
The ordered $k$-tuple $(L_1,\dots,L_k)$ is a {\em subsequence of} $\cvec c$
if for some increasing sequence of
indices $1\le j_1c$ or $a>bc_{i-2}$ and $c_{i-1}p$ and $j+p\le\len(\cvec c)$ is
recurrent. \item A class $\sC_\lrfs$ (and $\lrfs$) is {\em
recurrent} if $\lrf$ is recurrent for every $\lrf\in\lrfs$.
\end{itemize}
\end{definition}
\noindent It is a consequence of these definitions that if $\cvec
r$ and $\cvec s$ are recurrent subcompositions, $\len(\cvec a)\ge
p$, $\len(\cvec z)\ge p$, and $\cvec a\,\cvec x\,\cvec z\in\sC$,
then there is a composition $\cvec a\cdots\cvec r\cdots\cvec
s\cdots\cvec z$ in $\sC_\lrf$. (We get $\cvec a\cdots\cvec r\cvec
y\cvec z$ for some $\cvec y$. Replace $\cvec a$ with $\cvec
a\cdots\cvec r$ and $\cvec x$ with $\cvec y$ in the definition.)
For the first $\lrf$ in Example~\ref{ex:up-down}, the 2-part
subcomposition $ab$ is recurrent at 1~modulo~2 whenever $0y$ weak.
However, weak and strong inequalities give restrictions
which are not similar.
For example, if the restrictions in $\lrf'$ were changed to
weak giving $\lrf''$, we would not have $\lrf\approx\lrf''$
and so we could not apply Theorem~\ref{thm:main} to
$\{\lrf,\lrf''\}$.
\EndExample
\BeginRemark[Some asymptotics]
We refer to (a) and (b) near the start of this section.
Since the radius of convergence $r$ in (a) depends only on
the recurrent subcompositions, it will follow that the form
$A(r^{-n}(1+O(\delta^n))$ still holds for $\sC_\lrfs$
when $\lrfs$ is a {\em finite} set of similar restrictions.
For essentially the same reason, the normality in (b) continues
to hold.
(See Section~\ref{sec:LRC2} for details.)
\EndRemark
\begin{definition}
[Asymptotically free]
\label{def:asymp free}
Let $\sC_\lrf$ be a set of locally restricted compositions
of span $p$.
If $\sC_\lrf$ is recurrent and
the following hold, we say that $\lrf$ and the compositions in
$\sC_\lrf$ are {\em asymptotically free.}
\begin{itemize}
\item[(a)]
Suppose $j$ and $r_i$ are such that
$\cvec r(x)=r_1\cdots r_pxr_{p+2}\cdots r_{2p+1}$
is recurrent at $j$ modulo $m$ for infinitely
many values of $x$.
Then there is an $M$ (depending on $j$ and the $r_i$)
such that, if $\cvec r(x)$ occurs at a position $j$
mod~$m$ in a composition, we may replace that $x$
by any $x'\ge M$.
\item[(b)]
There is at least one set of values $j$ and $r_i$ of the sort
described in (a).
\end{itemize}
Let $\lrfs$ be a {\em finite set of similar} local restriction functions.
If $\lrf\approx\lrf'$ and $\lrf$ is asymptotically free, then
clearly $\lrf'$ is asymptotically free.
Hence we say that $\lrfs$ and the compositions in $\sC_\lrfs$ are
{\em asymptotically free} if $\sC_\lrf$
is asymptotically free for some $\lrf\in\lrfs$.
\end{definition}
\noindent
It is fairly easy to verify that asymptotically free
$\sC_\lrf$ are special cases of the regular $\sC_\lrf$ studied in~\cite{BC2}.
Note that, since $\lrf$ has span $p$, no parts other than the $r_i$ impose
restrictions on $x$.
We arrived at the notion of asymptotically free as a concept succinctly
stated, fairly intuitive, and inclusive of a number of known examples, for
which the results of Theorem~\ref{thm:main} hold.
It would be of interest to extend these results to more classes of compositions.
\BeginExample[A bad definition]
We could have attempted to define asymptotically free $\lrfs$ by simply insisting
that~(a) and~(b) hold for $\sC_\lrfs$, however this is insufficient.
Consider $\lrfs=\{\lrf,\lrf'\}$ and $\lrf$ (resp.\ $\lrf'$) requires that parts in
odd (resp.\ even) positions be odd.
Then large odd parts will tend to be more common than large even parts and so the
conclusion in Theorem~\ref{thm:main}(a) would be false.
\EndExample
\BeginExample[Generalized Carlitz compositions]
Carlitz compositions are defined by the restriction $c_i\ne c_{i-1}$.
They were generalized to restricted differences in~\cite{BC1} by requiring that
$c_i-c_{i-1}\notin{\cal N}$ where $\cal N$ is a fixed set of integers.
(Carlitz compositions correspond to ${\cal N}=\{0\}$.)
These compositions are recurrent with modulus~1 and span~1.
If $\cal N$ is finite, we have asymptotically free compositions.
For the generalized Carlitz compositions studied in \cite{BC1},
$\cal N$ was the same for all $c_{i-1}$. We can generalize further
by letting $\cal N$ depend on the value of $c_{i-1}$, say
${\cal N}(c_{i-1})$. If all the ${\cal N}(c)$ are finite, we still have
asymptotically free compositions; however, they cannot be studied
by the method in~\cite{BC1}. Instead,~\cite{BC2} must be used.
\EndExample
\BeginExample[Some periodic conditions]
Up-down compositions have constraints of modulus~2.
General periodic inequality constraints were studied in~\cite{BC3}.
These are all asymptotically free provided they allow parts to
both increase and decrease.
As in the preceding example, we could require that the change between
adjacent parts be dependent on the parts. For example, we could require
that the ratio of adjacent parts be at least~2 ($c_i/c_{i-1}\ge2$ for an
increase and $c_{i-1}/c_i\ge2$ for a decrease).
For fixed $k$, $k$-rowed compositions $a_{i,j}$ in which
differences of adjacent parts avoid a finite set are
asymptotically free.
One interleaves the parts to produce a one-rowed composition:
If $a_{i,j}$ are the parts of a $k$-rowed composition of $n$, then
$c_{i+k(j-1)}=a_{i,j}$ for $1\le i\le k$ and
$j=1,2,\cdots$ gives a bijection with one-rowed compositions
$\cvec c$ of $n$.
We can take the modulus and span to be $k$.
\EndExample
\begin{definition}
[Gap free]\label{def:gap-free}
A composition with largest part $M$ is called {\em gap free} if it
contains all recurrent parts less than $M$.
\end{definition}
\noindent
The restriction of gap-free to recurrent parts is used to rule out
classes such as the following. Let $\sC$ be all compositions subject
to the restriction that 2 and 3 can appear only as the first
part of a composition. Since almost all compositions contain 1 and
no composition in $\sC$ can contain both 2 and 3, almost no
compositions in $\sC$ would be gap-free if we required that the
support of the parts be an interval in $\naturals$.
\begin{conventions}
We use the following conventions in this paper.
\begin{itemize}
\item
When we talk about something random, we always mean that it
is chosen uniformly at random from the set in question. We say that a property
holds {\em asymptotically almost surely} (a.a.s) if the probability that the property holds
tends to 1 as the size of the set goes to infinity, and we also say that
{\em almost all} objects in the set have the property.
\item
Expectation is denoted by $\Ex$.
\item
After a class of compositions has been defined, we usually omit
the modifiers (e.g.\ asymptotically free) and refer to elements of the
class simply as compositions.
\item
The number of compositions of $n$ in the class $\sC$ is
asymptotically $Ar^{-n}$.
We will always use $A$ and $r$ for these parameters.
\item
All logarithms are to the base $1/r$ except the natural logarithm $\ln$.
\end{itemize}
\end{conventions}
Remember that we call $\sC_\lrfs$ asymptotically free if and only if $\lrfs$
is a finite set of similar asymptotically free local restriction functions.
\begin{theorem}[Main theorem]\label{thm:main}
Let $\gamma\doteq 0.577216$ be Euler's constant and let
\begin{equation}\label{eq:oscillation}
P_k(x) ~= ~\log e\sum_{\ell\ne 0}
\Gamma(k+2i\pi \ell\log e)\exp(-2i\ell\pi \log x).
\end{equation}
(This is a periodic function of $\log x$.
For $1/20$, which has the same value
in all parts of the theorem.
\begin{itemize}
\item[(a)]
Select a composition of $n$ uniformly at random.
Let $X_0(n)$ be the number of parts and $X_k(n)$ the number of parts
of size $k$.
For recurrent $k$ and $\epsilon>0$,
\begin{equation}\label{eq:Xk/X0}
{\rm Prob}\left(
\left|
\frac{X_k(n)}{X_0(n)} - \frac{\Ex(X_k(n))}{\Ex(X_0(n))}
\right|
> \epsilon
\right)
\to 0
~~\mbox{as $n\to\infty$}.
\end{equation}
Furthermore, the limit
\begin{equation}\label{eq:Ex/Ex}
\p_k = \lim_{n\rightarrow\infty}
\frac{\Ex(X_k(n))}{\Ex(X_0(n))}
\end{equation}
exists, and $\p_k \sim Br^k$ as $k\to\infty$ for some positive
constant $B$.
\item[(b)]
Let the random variable $M_n$ be the size
of the maximum part in a random composition of $n$. For any
function $\omega_b(n)$ such that $\omega_b(n)\to\infty$ as
$n\to\infty$, $|M_n-\log n|<\omega_b(n)$ a.a.s. Furthermore
$$
\Ex(M_n) ~=~
\log\left(\frac{Cn}{1-r}\right)+\gamma\log e
- \frac{1}{2}+P_0\!\left(\frac{Cn}{1-r}\right)+o(1),
$$
where $C = B\lim_{n\to\infty}\Ex(X_0(n))/n$.
\item[(c)]
Let $\nu$ be the number of nonrecurrent parts.
(Since the compositions are asymptotically free, $\nu$ is
finite.) Let the random variable
$D_n$ be the number of distinct $\underline{\mbox{recurrent}}$
parts in a random composition of $n$. For any function
$\omega_c(n)$ such that $\omega_c(n)\to\infty$ as $n\to\infty$,
$|D_n-\log n|<\omega_c(n)$ a.a.s. Furthermore
$$
\Ex(D_n) + \nu ~=~
\log(Cn)+\gamma\log e-\frac{1}{2}+P_0(Cn)+o(1).
$$
\item[(d)]
Let $q_n(k)$ be the fraction of compositions of $n$ which are
gap-free and have largest part $k$.
There is a function $\omega_d(n)\to\infty$ as $n\to\infty$ such that
\begin{equation}\label{eq:qn(k)}
q_n(k) ~\sim~
\exp\left(\frac{-Cnr^{k+1}}{1-r}\right)
\prod_{j\le k}\left(1-\exp\left(-Cnr^j\right)\right)
\end{equation}
uniformly for $|k-\log n|<\omega_d(n)$. Furthermore, for any
constant $D$, the minimum of $q_n(k)$ over $|k-\log n|0$.\cr}
\end{equation}
\item[(f)]
Let $g_n(k)$ be the fraction of compositions of $n$ that have exactly
$k$ parts of maximum size.
Then for each fixed $k$ and as $n\to\infty$,
$$
g_n(k) ~\sim~
\frac{(1-r)^k}{k!}P_k\left(\frac{Cn}{1-r}\right)+\frac{(1-r)^k\log
e}{k}.
$$
\item[(g)]
Let $D_n(k)$ be the number of distinct recurrent parts that appear
exactly $k$ times in a random composition of $n$. For fixed $k>0$
$$
\Ex(D_n(k)) ~=~ \frac{P_k(Cn)}{k!} + \frac{\log e}{k} + o(1).
$$
Let $m_n(k)$ be the probability that a randomly chosen recurrent part size
in a random composition of $n$ has multiplicity $k$.
For fixed $k$, $m_n(k)\sim \Ex(D_n(k))/\log n$.
\item[(h)]
Let $\lrfs'$ be a finite set of local restriction functions similar to
those in $\lrfs$.
The values of $r$, $B$ and $C$ are the same for $\sC_\lrfs$ and
$\sC_{\lrfs'}$.
\end{itemize}
\end{theorem}
\noindent
We recall that $\Gamma(a+iy)$ goes to zero exponentially fast as $y\to\pm\infty$.
Thus the sum (\ref{eq:oscillation}) is dominated by the terms with small $\ell$.
Parts (b) and (c) of the theorem can be thought of in terms of the weak law of
large numbers.
For example, (b) tells us that, for all $\eps>0$,
$$
\Pr\left(\left|\frac{M_n}{\Ex(M_n)}-1\right|>\eps\right) \to 0
~~{\rm as}~~ n\to\infty,
$$
and the condition on $\omega_b(n)$ provides a bound on the rate of convergence.
\medskip
Since estimating $C$ is generally harder than estimating $A$,
the following theorem is sometimes useful.
\begin{theorem}[Sometimes $A=C$]\label{thm:AeqC}
Let $\sC$ be a class of asymptotically free compositions
and let the number of compositions of $n$ be asymptotic
to $Ar^{-n}$.
Suppose that there is some $\ell$ such that,
whenever the number of parts in each of $\cvec a$ and
$\cvec b$ is at least $\ell$, we have that
$\cvec c=\cvec a x\cvec b$ is in $\sC$ for
infinitely many $x$ if and only if $\cvec a$ and $\cvec b$
are in $\sC$.
Then $C=A$, where $C$ is the constant in Theorem~\ref{thm:main}.
\end{theorem}
\begin{theorem}[Asymptotically Poisson]\label{thm:Poisson}
Let $\zeta_j$ be the number of parts of size $j$ in a random
composition in $\sC$ of size $n$. Then there is a function
$\omega(n)\rightarrow\infty$ such that the random variables
$\{\zeta_j:\log n -\omega(n) \le j\le n\}$ are asymptotically
independent Poisson random variables with means $\mu_j =Cnr^j$.
\end{theorem}
\section{Discussion and Examples}
\BeginRemark
[Some previous results]
We review some results that involve the study of parts of large size.
Most results deal with unrestricted compositions.
As far as we know, the first result is due to Odlyzko and Richmond~\cite{OR}.
For $a(n,m)$, the number of compositions of $n$ with largest part $m$,
they prove the sequence is unimodal for each $n$ and show that the $m$
which maximizes $a(n,m)$ is always one of the two integers closest to $\log_2 n$.
The fact that the largest part $M_n$ is strongly concentrated is well known.
For example, it appears as an exercise in~\cite{FS}.
Hwang and Yeh~\cite{HY} studied the distinct parts in a random composition,
obtaining asymptotics for the expected value of their number and sum as well
as other results.
Hitczenko and Stengle~\cite{HS} also studied the expected number of distinct parts.
The asymptotic probability that a composition is gap-free was
obtained by Hitczenko and Knopfmacher~\cite{HK}.
They based their proof on a gap free result they obtained for samples of iid
geometric random variables, which we also use in our study of gap-freeness.
Wilf asked about $m_n(k)$, the probability that a randomly chosen part size in a
random composition of $n$ had multiplicity $k$.
This problem was studied by Hitczenko, Rousseau and Savage~\cite{HSv,HRS}.
Louchard~\cite{Lo} studied $D_n(k)$, obtaining information about its moments.
Archibald and Knopfmacher~\cite{AK} studied the largest missing part in
compositions that are not gap free.
Fewer results have been obtained for Carlitz compositions.
Using~\cite{FGD}, Knopfmacher and Prodinger~\cite{KP} obtained asymptotics
for the largest part in Carlitz compositions and observed that there was
oscillatory behavior.
The expected number of distinct parts, $\Ex(D_n)$, was studied by
Hitczenko and Louchard~\cite{HL} who required an independence assumption
that was eliminated by Goh and Hitczenko~\cite{GH}.
Kheyfets~\cite{Khe} obtains results for parts of multiplicity $k$ that
parallel those mentioned in the previous paragraph for $D_n(k)$ and
$m_n(k)$ in the unrestricted case.
Louchard and Prodinger~\cite{LP} study the distribution of part sizes.
Theorem~\ref{thm:main} extends most of these results to asymptotically free compositions.
One exception is~\cite{AK} which came to our attention when this paper was
essentially complete.
It is likely that our methods can generalize their results, although with less
accuracy than they obtain.
Most of the known results for unrestricted and Carlitz compositions have
greater accuracy than our results which typically have $o(1)$ error rather
than more explicit estimates.
Also, we do not have formulas for the two constants $C$ and $r$ appearing in
our results, whereas they are known for unrestricted and Carlitz compositions.
However, since the number of compositions is $Ar^{-n}$ with an exponentially
small relative error the more important $r$ is easily estimated if one can
count compositions for moderate values of $n$ efficiently.
An earlier version of this paper appeared, without proofs, as the extended
abstract~\cite{BCG}.
The present paper considers a more general class of compositions and
contains some additional results.
\EndRemark
\BeginExample[$A=C$]
It is easily seen that Theorem~\ref{thm:AeqC} applies to the following
classes of compositions
\begin{itemize}
\item[(a)]
unrestricted compositions (so $C=1/2$);
\item[(b)]
compositions where the value of $c_i$ is restricted only by
$c_{i-1}$ and $c_{i+1}$ and may be arbitrarily large;
\item[(c)]
alternating compositions ($c_{2i-1}c_{2i+1}$)
where the number of parts must be odd.
\end{itemize}
We note that (b) includes Carlitz compositions and so
$C\doteq 0.4563634741$ for Carlitz compositions~\cite{LP}.
The inequality conditions in (c) can be generalized:
we may require that $c_{2i}-c_{2i-1}$ and $c_{2i}-c_{2i+1}$ belong
to some subset of $\integers$ that contains arbitrarily large
positive values and the subset may depend on $i$ modulo some period.
Although (c) gives $A=C$ for only one type of alternating compositions,
it follows from Theorem~\ref{thm:main}(h) that the value obtained for~$r$,
$B$ and~$C$ in this case are the same for the various types of alternating
compositions discussed in Example~\ref{ex:up-down} even though they
have differing values of~$A$.
\EndExample
\BeginExample[Gap-free]\label{ex:gap-free}
The numbers $p_m$ in (e) were studied by Hitczenko and Knopfmacher~\cite{HK}
who showed that they oscillated with the same period as~(\ref{eq:oscillation})
when $r>1/2$.
They showed that, for $r=1/2$, there is no oscillation.
Their Figure~7 shows that the amplitude of oscillation of $p_m$ is less
than $10^{-6}$.
Consequently, if $r$ is known, one can determine the asymptotic value of $p_m$
and hence $q_n$ to within $10^{-6}$.
The following are the values of $p_m$ for three families of
compositions, correct up to the sixth decimal place.
\begin{itemize}
\item
For Carlitz compositions, it is known $r\doteq .57134979$.
It follows from (\ref{eq:pm}) that $p_m\doteq 0.372000$ for $m\ge 25$.
\item
For strictly alternating compositions ($c_{2i-1}c_{2i+1}$),
$r\doteq 0.63628175$ by~\cite{BC3}.
It follows from (\ref{eq:pm}) that $p_m\doteq 0.252277$ for $m\ge 25$.
\item
For weakly alternating compositions ($c_{2i-1}\le c_{2i}\ge c_{2i+1}$),
$r\doteq .57614877$ by~\cite{BC3}.
It follows from (\ref{eq:pm}) that $p_m\doteq 0.363144$ for $m\ge 25$.
\end{itemize}
Here is an alternative definition of gap-free based on the literature:
A composition is gap-free if, whenever it contains two recurrent parts,
say $a$ and $b$, it contains all recurrent parts between $a$ and $b$.
This definition does not alter the conclusions of
Theorem~\ref{thm:main}(d,e) because, by Lemma~\ref{lemma:normality}(b)
below, the fraction of compositions of $n$ that omit the smallest recurrent
part is exponentially small.
\EndExample
\BeginExample[Conjectures of Jakli\v c, Vitrih and \v Zagar]
Let ${\rm Max}_k(n)$ (resp.\ ${\rm Min}_k(n)$)
denote the number of all compositions of $n$ such that there are
more than $k$ copies of the maximal (resp.\ minimal) part.
Jakli\v c et al.~\cite{JVZ} conjectured that, when $k=1$
\begin{eqnarray}
& & \lim_{n\to \infty} \frac{{\rm Min_k}(n+1)}{{\rm Min_k}(n)} ~=~ 2
\label{eq:ConjMin}\\
& & \lim_{n\to \infty} \frac{{\rm Max_k}(n+1)}{{\rm Max_k}(n)} ~=~ 2.
\label{eq:ConjMax}
\end{eqnarray}
In fact, the conjectures hold for the compositions studied in this paper and
all $k\ge1$ provided 2 is replaced with $1/r$ and Min is restricted to recurrent parts.
The number of occurrences of any given recurrent part is $\Theta(n)$
for almost all recurrent locally restricted compositions of $n$ by~\cite{BC2}.
Thus (\ref{eq:ConjMin}) follows immediately from the fact that the number of
compositions of $n$ is asymptotic to $Ar^{-n}$.
We now prove (\ref{eq:ConjMax}).
Note that
$$
{\rm Max}_k(n) ~\sim~ Ar^{-n}\Biggl(1-\sum_{i\le k}g_n(i)\Biggr).
$$
By Theorem~\ref{thm:main}(f), $g_n(i)\sim g_{n+1}(i)$ and
$g_n(i)$ is bounded away from zero as $n\to\infty$.
Thus
$$
\frac{{\rm Max_k}(n+1)}{{\rm Max_k}(n)}
~\sim~
\frac{\Bigl(1-\sum_{i\le k}g_{n+1}(i)\Bigr)Ar^{-n-1}}
{\Bigl(1-\sum_{i\le k}g_n(i)\Bigr)Ar^{-n}}
~\sim~
\frac{1-\sum_{i\le k}g_{n+1}(i)}{1-\sum_{i\le k}g_n(i)}
\frac{1}{r}
~\sim~ \frac{1}{r}.
$$
One can change the definition of ${\rm Max}_k$ to mean exactly
$k$ copies of the maximal part and a similar proof will hold.
\EndExample
\BeginExample[Counterexamples without freeness]
It was shown in Theorem~1(f) of~\cite{BC1} that when differences of adjacent
parts are restricted to a finite set, the largest part is asymptotically almost surely of
order $\sqrt{\log n}$, so the bound in Theorem~\ref{thm:main}(a) fails.
\EndExample
\section{Statement of Lemmas}
The following six lemmas are used in our proofs of Theorems~
\ref{thm:main}, \ref{thm:AeqC} and~\ref{thm:Poisson}.
\begin{lemma}[Normality and tails]\label{lemma:normality}
Let $\sC_\lrfs$ be a class of asymptotically free compositions
and let $d$ be arbitrary.
Let ${\cal R}$ be a possibly infinite nonempty set of recurrent
subcompositions each of which contains at most $d$ parts.
Assume that if we alter $\lrfs$ to forbid the elements of ${\cal R}$,
the resulting class of compositions is still recurrent.
Let the random variable $X_n$ be either the number of
occurrences of elements of $\cal R$ in a random composition of $n$
or the number of parts in a random composition of $n$.
The following are true.
\begin{itemize}
\item[(a)]
The distribution of $X_n$ is asymptotically normal with mean and
variance asymptotically proportional to $n$.
\item[(b)]
There are constants $C_i>0$ depending on what $X_n$ counts such that
$$
\Pr(X_n\!<\!C_1n) ~<~ C_2(1+C_3)^{-n}
~~\mbox{for all $n$.}
$$
\item[(c)]
Let $\cvec s$ be a subcomposition.
There is a constant $B$ dependent only on $\sC$ such that the
probability that a random composition contains at least one copy
of $\cvec s$ is at most $Bnr^{\partsum(\cvec s)}$.
\end{itemize}
\end{lemma}
\begin{definition}[The function $\free$]
\label{def:free}
As in Definition~\ref{def:asymp free} let $\cvec r(x)=r_1\cdots r_pxr_{p+2}\cdots r_{2p+1}$
where $\cvec r=\cvec r(0)=r_1\cdots r_p0r_{p+2}\cdots r_{2p+1}$. For an
asymptotically free class the set
$$
S(\cvec r) = \{x: \cvec r(x) {\rm ~is~recurrent~}\}
$$
is either finite or co-finite. So, there is a smallest integer $q(\cvec r)$ such that
$$
{\rm either~} [q(\cvec r),\infty) \subseteq S(\cvec r)
{\rm ~~or~~}
[q(\cvec r),\infty) \subseteq \overline{S(\cvec r)}.
$$
Define
$$
\free(P) = \max\{q(\cvec r): r_i \le P {\rm ~for~} 1\le i\le 2p+1\}.
$$
\end{definition}
\noindent
It follows from the definition that if $\max(\cvec r)\le P$
and $x\ge\free(P)$ and $\cvec r(x)$ is recurrent at $j$ modulo $m$,
then $\cvec r$ is asymptotically free at $j$ modulo $m$.
\begin{definition}[$P$-isolated]
Suppose $\cvec c= c_{i-p}\cdots c_i\cdots c_{i+p}$
is a recurrent subcomposition.
If no $c_j$, except possibly $c_i$, exceeds $P$,
we call $c_i$ {\em $P$-isolated}.
\end{definition}
\noindent
A consequence of the definitions is that, whenever $x\ge\free(P)$
is $P$-isolated we are free to replace $x$ by any part that is of
size $\free(P)$ or greater.
\begin{lemma}[Large part separation]\label{lemma:separated}
Let $\sC_\lrfs$ be a class of asymptotically free compositions.
Suppose $\delta>0$.
There is a $P=P(\delta)$ and $N=N(\delta)$
such that the following holds for every $m\ge\free(P)$.
Let ${\cal M}(n)$ be the set of compositions of $n$ in which a part
of size $m$ has been marked.
For all $n>N+m$ the subset of ${\cal M}(n)$ in which the marked part
is not $P$-isolated has size less than $\delta|{\cal M}(n)|$.
\end{lemma}
\noindent
The following lemma proves most of Theorem~\ref{thm:main}(a).
\begin{lemma}[Geometric probabilities]\label{lemma:geometric}
We use the notation of Theorem~\ref{thm:main}(a).
\begin{itemize}
\item[(a)]
Equations (\ref{eq:Xk/X0}) and (\ref{eq:Ex/Ex}) are true.
\item[(b)] Recall that $\p_k$ is the limit (on $n$)
of the ratio $\Ex(X_k(n))/\Ex(X_0(n))$.
For all sufficiently large parts $k$ and $\ell$ depending on $\delta>0$,
we have
$$
\left|\frac{\p_k\; r^{-k}}{\p_\ell\; r^{-\ell}}- 1\right| ~<~ \delta.
$$
\item[(c)]
We have $\p_k \sim Br^k$ for some positive constant $B$.
\end{itemize}
\end{lemma}
\begin{lemma}[Marked compositions]\label{lemma:L count}
Fix $k$ and a class $\sC$ of asymptotically free compositions.
Let $A$ be such that the number of compositions of $n$ is asymptotic to
$Ar^{-n}$ and let $C$ be as in Theorem~\ref{thm:main}.
If $\L(n)=(L_1(n),\ldots,L_k(n))$ is a sequence of $k$-tuples of
integers with
$$
\max(L_i)=o(n) ~~\mbox{and}~~ \min(L_i)\to\infty
~\mbox{as}~n\to\infty,
$$
then the number of compositions $\cvec c$ of $n$ having
$\L=(L_1 \cdots L_k)$ as a marked subsequence is
\begin{equation}\label{eq:L count}
(A+o(1))\frac{(Cn)^k r^{s-n}}{k!}
~\mbox{as}~ n\to\infty,
~\mbox{where}~ s=L_1+\cdots+L_k.
\end{equation}
\end{lemma}
\begin{lemma}[Characterization of Poisson]\label{GW12}
Let $(m)_k:=m(m-1)\cdots (m-k+1)$
denote the falling factorial.
Suppose that $\zeta_1, \ldots, \zeta_n =\zeta_1(n), \ldots,
\zeta_n(n)$ is a set of non-negative integer variables on a
probability space $\Lambda_n$, $n=1,2, \ldots,$ and there is a
sequence of positive reals $\gamma(n)$ and
constants $0<\alpha<1$ and $00)=O\left(c^{k(n)-\gamma(n)}\right)$ uniformly
for all $k(n)>\gamma(n)$.
\end{description}
Then there exists a function $\omega(n)\to\infty$ so that
for $k=\lfloor\gamma(n)-\omega(n)\rfloor$, the total variation distance between
the distribution of $(\zeta_k,\zeta_{k+1},\ldots,\zeta_n)$, and that of
$(Z_k,Z_{k+1},\ldots,Z_n)$ tends to 0, where the $Z_j = Z_j(n)$ are
independent Poisson random variables with $\Ex Z_j = \alpha^{j-\gamma(n)}$.
\end{lemma}
\BeginRemark
The preceding lemma, Lemma~\ref{GW12}, is applied to obtain the Poisson result for large parts
stated as Theorem~\ref{thm:Poisson}. The latter, in turn,
is used with Mellin transforms to prove Theorem~\ref{thm:main}(b-d);
and, with a result of Hitczenko and Knopfmacher~\cite{HK} on sequences of
geometric i.i.d.\ random variables, to prove Theorem~\ref{thm:main}(e,f).
\EndRemark
\begin{lemma}[Plentitude of recurrent parts]\label{lemma:zeta sum}
Let $\zeta_j$ be the number of occurrences of $j$ in
a random composition of $n$, and
let $k>0$ be arbitrary and fixed.
If $\omega(n)\to\infty$, then
$$
\sum_{\textstyle{j<\log n-\omega(n) \atop j{\rm ~recurrent}}}
\hskip-12pt
\Pr(\zeta_j\!<\!k) ~=~ o(1).
$$
\end{lemma}
\section{The Transfer Matrix and Sets of Functions}\label{sec:LRC2}
Before embarking on the proofs, we summarize some facts from~\cite{BC2}
which will be used and reduce the study of a finite set $\lrfs$ to a
single $\lrf$ since only single $\lrf$'s were considered in~\cite{BC2}.
We may replace the span $p$ by any larger value without altering the set
of compositions, provided we adjust the definition of $\lrf$.
Thus {\em we will assume that the span is a multiple of the modulus $m$.}
(Refer back to Definition~\ref{definition:LRF} for terminology.)
Let $\NC(n)$ be the number of compositions of $n$ in a regular,
locally restricted class $\sC_\lrf$, and let $F(x)=\sum\NC(n)x^n$ be the ogf
(ordinary generating function).
Then, as proven in Theorem 2 of
that paper,
\begin{equation}\label{eq:LR2gf}
F(x^2) = \varphi(x) + F_{NR}(x^2),
\end{equation}
where
\begin{equation}\label{eq:varphi}
\varphi(x)= {\bf s}(x)^{\Tr} \, \Biggl( \sum_{k=0}^{\infty}T(x)^k \Biggr) \, {\bf f}(x).
\end{equation}
We now explain the various parts of~(\ref{eq:LR2gf}).
Here a ``small number of parts'' is at most some small multiple of $p$.
\noindent{\bf The transfer matrix $T(x)$}
is defined in terms of a certain sequence of
words $\cvec\nu_1, \cvec\nu_2, \dots$, where by a {\it word}
we mean a recurrent subcomposition of length $p$, the span,
(see Definition~\ref{definition:LRF}) whose first part is at j~mod~$p$
where $j$ is the same for all words indexing $T$ (and thus ${\bf s}$
and ${\bf f}$ as well).
The list contains all such recurrent words and
$$
T(x)_{ij} ~=~ \cases{
x^{\Sigma(\cvec\nu_i)+\Sigma(\cvec\nu_j)} &
if $\cvec\nu_j$ can follow $\cvec\nu_i$,\cr
0 & otherwise.}
$$
Except for parts near the ends, every composition is a concatenation of such words.
A single application of the transfer matrix corresponds to the adjunction of $p$
additional parts to the composition.
\noindent{\bf The infinite vectors ${\bf s}(x),{\bf f}(x)$}
have analytic entries corresponding to compositions with a small number of parts.
The component $s_i(x)$ of ${\bf s}(x)$ deals with the generating function for
the beginning of compositions where the last $p$ parts in the beginning are $\cvec\nu_i$.
Similarly, $f_j(x)$ deals with the generating function for endings whose first
$p$ parts are $\cvec\nu_j$.
\noindent{\bf The function $F_{NR}(x)$}
is the ogf for the subclass of compositions not counted in $\varphi(x)$.
These compositions have at most some small number of parts.
The ogf $F_{NR}(x)$ has radius of convergence 1.
(This is slightly different from the definition of $F_{NR}$ in~\cite{BC2};
however, all that matters for the theory is that $F_{NR}$ has radius of
convergence~1 and that~(\ref{eq:LR2gf}) counts all compositions exactly once.)
To assure that $T(x)$ satisfies certain useful technical conditions, it is
necessary to have the arguments $x^2$ and $x$ as indicated in~(a).
See the latter part of this section and~\cite{BC2} for more details on $T(x)$.
\subsection{Reduction to a single $\lrf$ and Theorem~\ref{thm:main}(h)}
Before discussing the more technical issues related to asymptotics, we explain
why it suffices to consider one $\lrf$ instead of an entire finite set $\lrfs$ of
similar $\lrf$.
This discussion will also prove Theorem~\ref{thm:main}(h).
Suppose $\lrf\approx\lrf'$ and let $T$ be the transfer matrix for $\lrf$.
Since $T_{ij}\ne0$ if and only if $\nu_i\nu_j$ is recurrent, we can use the
same transfer matrix for $\lrf'$; however, the vectors ${\bf s}$ and ${\bf f}$
will be different.
In fact, if $k$ is as in Definition~\ref{def:similar}, the number of parts
in the subcompositions of the vectors ${\bf s}$ for $\lrf$ and $\lrf'$
will differ by $k$~mod~$m$.
Nearly all results in~\cite{BC2} depend on $T$ but not on ${\bf s}$ or ${\bf f}$.
The exception is the constant $A$ in the asymptotic estimate $Ar^{-n}$ for the
number of compositions of $n$.
It follows that, if the sets $\sC_\lrf$, $\lrf\in\lrfs$, were pairwise disjoint
we could simply obtain results for one $\lrf\in\lrfs$ and combine the results where,
whenever $A$ is present we simply sum the values of $A$ for the various
$\lrf\in\lrfs$.
We now show that this can, in principle, be done.
There is no need to do this in practice since analytic methods for obtaining reasonable
estimates of $A$ are seldom available even for a single $\lrf$.
Fix temporarily a $\lrf\in\lrfs$.
Partition the elements $\lrf'$ of $\lrfs$ into $m$ sets $\lrfs_0,\ldots,\lrfs_{m-1}$
according to the value of $k$ in Definition~\ref{def:similar}.
We now focus on these sets,
first considering functions in different sets and then functions in the same set.
Suppose $\lrf\in\lrfs_i$ and $\lrf'\in\lrfs_j$ where $i\ne j$.
Consider the compositions in $\sC_\lrf\cap\sC_{\lrf'}$.
Let the value of $\lrf\lrf'$ be simply the product of $\lrf$ and $\lrf'$.
We note that $\sC_\lrf\cap\sC_{\lrf'}=\sC_{\lrf\lrf'}$ since a composition is in the
intersection if and only if it satisfies both local restriction functions.
Since $i\ne j$, it follows that the transfer matrix for the intersection will be
the same as that for $\lrf$ with some nonzero entries replaced by zeroes.
By Lemma~2(f) of~\cite{BC2} and the realization that the spectral radius
determines the growth rate (see below) it follows that the number of compositions
of $n$ in the intersection grows at an exponentially smaller rate than the number
in $\sC_\lrf$.
Hence, for asymptotic purposes, we may treat the $m$ sets $\sC_{\lrfs_i}$ as if
they are disjoint.
It follows that, except for Theorem~\ref{thm:AeqC}, we may assume we are dealing
with just one $\lrfs_i$.
We now consider a single $\lrfs_k$.
Suppose $\cvec c\in\sC_{\lrfs_k}$ is counted by (\ref{eq:varphi}).
We can write it in the form $\cvec a\cvec b\cvec z$, where $\cvec b$
is a sequence of words $\cvec\nu$ that index $T$, $\cvec s$ and $\cvec f$,
$\cvec a$ ends with one of these $\nu$ and $\cvec z$ starts with one of them.
By absorbing a recursive word or two in $\cvec a$ and $\cvec z$ if needed,
we can insure that the following two assumptions hold for some $\ell_i$.
\begin{itemize}
\item[(i)]
Since all compositions come from the same $\sC_{\lrfs_k}$ we can assume
$\len(\cvec a)=\ell_0$, the same value for all compositions in $\sC_{\lrfs_k}$.
\item[(ii)]
Since multiplication by $T$ adds $p$ parts to the compositions,
we can assume that the longest and shortest values of $\len(\cvec z)$,
say $\ell_1$ and $\ell_2$, differ by less than $p$.
\end{itemize}
It follows that each composition in $\sC_{\lrfs_k}$ with at least $\ell_0+\ell_1$
parts is counted by ${\bf s}T^k{\bf f}$ for some $k$ and has uniquely determined
$\cvec a$ and $\cvec z$.
We can limit attention to compositions with at least $\ell_0+\ell_1$ parts since
the generating function for those with fewer parts has radius of convergence at
least~1.
With each $\lrf_i\in\lrfs_k$ we associate two sets $\set S_i$ and $\set F_i$ as
follows.
$\cvec a\in\set S_i$ and $\cvec z\in\set F_i$ if and only if they satisfy
(i) and (ii) above and $\cvec a\cvec b\cvec z\in\sC_{\lrf_i}$ for some $\cvec z$.
The set $\set S_i$ determines $\bf s$ as follows.
If $\cvec a\in\set S_i$ ends with $\nu_j$, then a generating function obtained
from $\cvec a$ is added to $s_j$.
A similar construction holds for $\cvec z$ and $\bf f$.
Thus $\set S_i\times\set F_i$ determines the compositions in $\sC_{\lrf_i}$.
If we had $(\set S_i\times\set F_i)\cap(\set S_j\times\set F_j)=\emptyset$,
it would follow that $\sC_{\lrf_i}\cap\sC_{\lrf_j}$ would contain at most
some compositions shorter than $\ell_0+\ell_1$.
Thus, we need only prove that a union of Cartesian products
$\cup_{\lrf_i\in\lrfs_k}(\set S_i\times\set F_i)$
can always be written as a disjoint union of such products. This is done by
considering the given terms $\set S_i\times\set F_i$ one at the time, and
using the identity
$$
(A\times B) \cap (C \times D)^{c} ~=~ \Bigl(A\times(B\setminus D)\Bigr)
\cup \Bigl((A\setminus C)\times (B\cap D)\Bigr),
$$
where the union is disjoint.
(Think of $C\times D$ as the latest $\set S_i\times\set F_i$, and
$A\times B$ as one of the pairwise disjoint components of the
previously processed $(i-1)$ products. We keep $C\times D$ as a new
component, and each previously existing component is replaced by two
disjoint pieces.)
For each product in the resulting disjoint union, we construct a $\lrf$,
and their sum is the generating function for $\sC_k$, with the
possible exception of short compositions.
\subsection{Analytic aspects of $T(x)$ from~\cite{BC2}}
By Lemma 3 of~\cite{BC2}, at each $x_0\in(0,1)$
we have a neighborhood and functions $\lambda(x)$, $E(x$) and $B(x)$
analytic in that neighborhood such that
\begin{equation}
\label{eq:decomposition}
T(x)=\lambda(x) E(x)+B(x),
\end{equation}
where $E(x)$ is the projection onto the one-dimensional eigenspace
of eigenvalue $\lambda(x)$, the spectral radius of $B(x)$ is less than $\lambda(x)$,
and $E(x)B(x)=B(x)E(x)=0$. The proof of Lemma 3 relies heavily on results
and methods from \cite{Kato}.
If we choose for $x_0$ the point $r^{1/2}$, $0 r(1)
\end{equation}
for some $s$ and $\delta$.
By Lemma~2(f) of~\cite{BC2}, $\lambda(x,s)<\lambda(x,1)$ for $x>0$ and $0~~r(1)$ and so
(\ref{eq:lambda s}) holds for all sufficiently small $\delta$ depending on $s$.
This completes the proof of (b).
The proof of (c) is essentially the same as that given in
Section~9 of~\cite{BC2} for large part size. Since there are
slight changes, we repeat it here for completeness.
Let $p$ be the span of $\lrf$. Consider an expanded class
$\sC_\Psi$ where $\Psi$ is the same as $\lrf$ except that the
first $p$ and last $p$ parts of compositions are unrestricted.
The transition matrix $T(x)$ is unchanged.
Therefore $\Psi$ has the same radius of convergence $r$
as $\lrf$. Hence the number of compositions of $n$ in $\sC_\Psi$
is bounded above by $Cr^{-n}$ for some $C$. Hence the generating
function for $\sC_\Psi$ compositions by sum of parts is bounded
coefficient-wise by $C(1-x/r)^{-1}$.
Imagine marking a copy of $\cvec s$ in each composition in
$\sC_\lrf$. By the previous paragraph, the generating function for
such compositions of $n$ is bounded coefficient-wise by
$$
\frac{C}{1-x/r}x^s\frac{C}{1-x/r}
\mbox{~~where $s=\partsum(\cvec s)$.}
$$
Hence the number of such compositions of $n$ is bounded above by
$nC^2r^{s-n}$.
The previous paragraph overcounts the number of compositions
containing $\cvec s$. For some $C'>0$, the total number of
compositions of $n$ is at least $C'r^{-n}$ for large $n$. Taking
the ratio gives (c) with $B=C^2/C'$.
\EndProof
\section{Proof of Lemma~\ref{lemma:separated}}
Throughout the proof, whenever a new
condition is imposed on $P$ or $N$ it is understood that the
implied values must be at least as large as those already chosen.
All implied limits, as in $o(1)$, are as $n\to\infty$.
Let ${\cal M}^*(n)$ be the subset of ${\cal M}(n)$ in which the
marked part is not $P$-isolated.
Let $M^*(n)$ and $M(n)$ be the cardinalities of these two sets.
We will overestimate $M^*(n)$ and underestimate $M(n)$ and
show that their ratio can be made arbitrarily small provided $P$
and $n-m$ are sufficiently large.
For both counts, we consider compositions of the form
$\cvec a\cvec b\cvec c$ where $\cvec b$ contains a
special sequence of parts.
We then sum over $a$.
The composition $\cvec a$ will be like compositions in the
class $\sC_\lrf$ except that there will
be conditions on the last $p$ parts.
Since $T(x)$ is unchanged, the radius of convergence is
unchanged and so the number of $\cvec a$ is $\Theta(r^{-a})$
as $a\to\infty$.
A similar result holds for $\cvec c$.
We refer to this below as ``theta''.
Let $a=\partsum(\cvec a)$ and $c=\partsum(\cvec c)$.
We start with the underestimate of $M(n)$.
Let $\cvec r(x)$ be as in Definition~\ref{def:asymp free}
and let $b$ be the sum of its parts {\em excluding} $x$.
Choose $P$ so that $x$ is $P$-isolated in $\cvec r(x)$.
Thus $b$ is fixed as $n\to\infty$,
but we may increase $P$ as necessary later.
Since $m\ge\free(P)$, we may replace $x$ with a marked part $m$.
Let $\cvec b=\cvec r(m)$.
To underestimate $M(n)$, we will obtain a lower bound
on the number of occurrences of $\cvec b$.
Since the $r_i$ are fixed and the span is $p$,
the choices for $\cvec a$ and $\cvec c$ such that
$\cvec a\cvec b\cvec c\in\sC$ are independent of $m$.
By theta there are $B$ and $s$ such that there are at least
$Br^{-a}$ choices for $\cvec a$ and $Br^{-c}$ for $\cvec c$ when
$a\ge s$ and $c\ge s$.
Thus the total number of occurrences of $\cvec b$ is at least
$$
\sum_{a=s}^{n-(b+m)-s}B^2r^{-s}r^{-(n-s-(b+m))} ~=~
(n-2s-b-P+1)B^2r^{-n+b+m},
$$
and so for sufficiently large $n$ and some constant $C_0~~**0$ to be specified later, let
$\NC^+_j(n)$ be the number of those containing at least $\delta n$
copies of $\cvec r(x)$ and $\NC^-_j(n)$ be the remainder.
Let $\NC^-(n)$ count compositions with fewer than $\delta n$
copies of $\cvec r(x)$.
By Lemma~\ref{lemma:normality}(b) with $\delta=C_1$ sufficiently
small, $\NC^-(n)/\NC(n)$ goes to zero exponentially fast as $n\to\infty$.
Since $\NC_j^-(n)\le \NC^-(n)$, it follows that
$\NC_j^-(n)< C_2(1+C_3)^{-n}\NC(n)$ where the constants do not
depend on $j$.
In each composition counted by $\NC^+_j(n)$ replace $x$ by
$j$ in $k$ of the $\cvec r(x)$.
This can be done in at least ${\delta n\choose k}$ ways,
giving a composition of $n+k(j-x)$.
Since the resulting composition can have at most $2k-1$ parts of
size $j$, it could have arisen by this replacement process in at
most ${2k-1\choose k}$ ways.
Thus
$$
\NC^+_j(n){\delta n\choose k} ~\le~ \NC(n+k(j-x)){2k-1\choose k}
$$
and so
$$
\frac{\NC^+_j(n)}{\NC(n)}
~\le~ \frac{{2k-1\choose k}}{{\delta n\choose k}}\;
\frac{\NC(n+k(j-x))}{\NC(n)}
< \frac{Br^{-kj}}{n^k}
$$
for some $B=B(k,\delta,x)$ independent of $j$.
Thus
$$
\NC_j(n)/\NC(n) ~<~ Br^{-kj}/n^k + C_2(1+C_3)^{-n},
$$
where all constants are independent of $n$ and $j$.
Summing the right side over $Q\le j\le\log n-\omega(n)$,
we obtain the bound
$$
C_4r^{k\omega(n)} + C_2(1+C_3)^{-n}\log n ~=~ o(1)
$$
for some constants $C_i$.
\EndProof
\section{Proofs of Theorems \ref{thm:AeqC} and \ref{thm:Poisson}}
\BeginProofOf{Theorem~\ref{thm:AeqC}}
There is an increasing function $A_\ell(k,P)$ with supremum $A$
such that the number of compositions of $n\ge k$ with end parts
at most $P$ is at least $A_\ell(k,P)r^{-n}$.
There is a decreasing function $A_u(k)$ with infimum $A$ such that the
number of compositions of $n\ge k$ is at most $A_u(k)r^{-n}$.
Using the construction $\cvec c=\cvec ax\cvec b$ in the statement
of Theorem~\ref{thm:AeqC} together with the idea and notation
in the above proof of Lemma~\ref{lemma:L count},
we construct a composition with one marked part.
If $k\le t\le n-k-Q$, the number of compositions $\cvec aQ\cvec b$
with $\partsum(\cvec a)=t$ and $\partsum(\cvec b)=n-t-Q$
is between $A_\ell(k,P)^2r^{Q-n}$ and $A_u(k)^2r^{Q-n}$.
Sum over all $t$ in the interval $k\le t\le n-k-Q$.
Let $k\to\infty$ sufficiently slowly with $n$.
This shows that $C_P$ in the proof of Lemma~\ref{lemma:L count}
satisfies
$$
A^2 ~=~
\left(\lim_{k\to\infty}A_\ell(k,P)\right)^2
\le~ C_PA ~\le~ \left(\lim_{k\to\infty}A_u(k)\right)^2 =~A^2.
$$
The theorem follows.
\EndProof
\BeginProofOf{Theorem~\ref{thm:Poisson}}
We will show that the three hypotheses (i)-(iii) of Lemma~\ref{GW12}
are satisfied with the choices $\gamma(n)=\log(Cn)$, $\alpha=r$, and
$c=r$. The first, (i), is obvious.
For (ii), let $\ell,m_1,\dots,m_{\ell}$ be fixed and let
$k_1(n)\omega_d(n)$ tends to zero.
Thus it suffices to prove~(\ref{eq:qn(k)}).
By Theorem~\ref{thm:Poisson} there is some $\omega(n)\to\infty$
such that
\begin{eqnarray*}
q_n(k)
&\sim& p(n)
\Biggl(\prod_{j=f(n)}^k\left(1-\exp\left(-r^{j-\log(Cn)}\right)\right)\Biggr)
\Biggl(\prod_{j=k+1}^n\exp\left(-r^{j-\log(Cn)}\right)\Biggr)\\
&=& p(n)
\Biggl(\prod_{j=f(n)}^k\left(1-\exp\left(-Cnr^j\right)\right)\Biggr)
\exp\Biggl(-Cn\sum_{j=k+1}^n r^j\Biggr),
\end{eqnarray*}
where $f(n)=\lfloor\log(Cn)-\omega(n)\rfloor$ and $p(n)$ is the probability
that a random composition of~$n$ contains all recurrent parts less than $f(n)$.
With a little calculation, we see that Theorem~\ref{thm:main}(d)
is equivalent to $p(n)\sim1$.
With $\zeta_j$ as in Lemma~\ref{lemma:zeta sum}, we have
$$
p(n) ~\ge~ 1-\sum_{j\le f(n)}\Pr(\zeta_j\!=\!0) ~=~ 1-o(1).
$$
We now prove the final claim about strong concentration.
Since the probability of being gap-free is bounded away from zero
by previous claim in (d), it suffices to prove the validity of the
statement for all compositions. The result now follows from (b).
This proves Theorem~\ref{thm:main}(d).
\EndProof
\BeginProofOf{Theorem~\ref{thm:main}(b,c)}
We first turn to the formula in (c).
Since $\zeta_j$ in Theorem~\ref{thm:Poisson} is irrelevant for small $j$,
we let $\zeta_j$ be as in Lemma~\ref{lemma:zeta sum} and
$\omega(n)$ be any function that goes to infinity.
Note that
$$
\Ex(D_n)+\nu ~=~ \sum_{j=1}^n \Pr(\zeta_j\ne 0)
$$
and so, by Lemma~\ref{lemma:zeta sum},
\begin{eqnarray}
\Ex(D_n)+\nu
&=& o(1) ~+\!\! \sum_{j=1}^{\kpn-1}\!\!1
\label{eq:Dn split 1}\\
&& +\!\! \sum_{j=\kmn}^{\kpn}\!\!\Pr(\zeta_j\ne 0)
\label{eq:Dn split 2}\\
&& +\!\! \sum_{j=\kpn+1}^n\!\!\Pr(\zeta_j\ne 0).
\label{eq:Dn split 3}
\end{eqnarray}
By Lemma~\ref{lemma:normality}(c)
$$
\Pr(\zeta_j\ne0) ~=~ O(nr^j)
~~\mbox{provided $j\to\infty$ with $n$.}
$$
Thus the sum in (\ref{eq:Dn split 3}) is
$O(r^{\omega(n)})=o(1)$.
By the Poisson distribution, the terms in the sum (\ref{eq:Dn split 2})
are $1-\exp\left(-Cnr^j\right)+o(1)$ and so, if $\omega(n)\to\infty$
sufficiently slowly, that sum is
$$
o(1)+\sum\Bigl(1-\exp\left(-Cnr^j\right)\Bigr).
$$
Since the sum of $\exp\left(-Cnr^j\right)$ over $j<\kmn$ is $o(1)$,
we may replace 1 in the sum (\ref{eq:Dn split 1}) with
$1-\exp\left(-Cnr^j\right)$.
Finally,
$$
\sum_{j>\kpn}\!\!\Bigl(1-\exp\left(-Cnr^j\right)\Bigr) ~=~ o(1)
$$
and so
$$
\Ex(D_n)+\nu
~=\! \sum_{j\ge 0}\Bigl(1-\exp\left(-Cnr^j\right)\Bigr)-1+o(1).
$$
Let $f(x) = \sum_{j\ge 0}\Bigl(1-\exp\left(-xr^j\right)\Bigr)$.
Then $\Ex(D_n)=f(Cn)-1+o(1)$.
We use the standard Mellin transform.
(See \cite[p.765]{FS}, and also their Example B.5 which
treats $r=1/2$).
It follows that
$$
f(x) ~=~ \log x+\gamma\log e+\frac{1}{2}+P_0(x)+o(1),
$$
where $P_0(x)$ is given
by~(\ref{eq:oscillation}).
This proves Theorem~\ref{thm:main}(c).
For the maximum part size $M_n$, we proceed in a similar manner:
\begin{eqnarray*}
\Ex(M_n)
&=&\sum_{j=1}^n\Pr(M_n\ge j)\\
&=&\!\sum_{j=1}^{\kmn-1}\!1
~+\!\!\! \sum_{j=\kmn}^{n}\!\Bigl(1-\Pr(\wedge_{i\ge j} \{\zeta_j=0\})\Bigr)+o(1)\\
&=&\sum_{j=1}^{\kmn-1}\hskip-12pt 1 ~~+\!\!
\sum_{j=\kmn}^{n}\!\!\left(1-\exp\left(-\frac{Cn}{1-r}r^j\right)\right)+o(1)\\
&=&\sum_{j\ge 0}^{n}\left(1-\exp\left(-\frac{Cn}{1-r}r^j\right)\right)-1+o(1)
=f\left(\frac{Cn}{1-r}\right)-1+o(1)\\
&=& \log\left(\frac{Cn}{1-r}\right) + \gamma\log e
- \frac{1}{2}+P_0\left(\frac{Cn}{1-r}\right)+o(1).
\end{eqnarray*}
It remains to prove the claims about $|D_n-\log n|$ and $|M_n-\log n|$.
Since $D_n\le M_n$, it suffices to establish the lower bound for $D_n$ and
the upper for $M_n$.
The upper bound on $M_n$ was proved in \cite[Section~9]{BC2}.
In the proof of Theorem~1(d) we showed that $p(n)\sim1$,
which establishes the lower bound on $D_n$.
\EndProof
\BeginProofOf{Theorem~\ref{thm:main}(e--g)}
Let $\Gamma:=(\Gamma_1,\Gamma_2,\ldots,\Gamma_m)$ be a
sequence of i.i.d.\ geometric random variables with parameter
$p=1-r$.
Hitczenko and Knopfmacher~\cite{HK} showed that the probability
the sequence $\Gamma$ is gap-free is given by the $p_m$
in our~(\ref{eq:pm}) and they established the oscillation of $p_m$
when $p\ne1/2$.
Let $\omega(m)$ go to infinity arbitrarily slowly with $m$.
Let $M_m$ be the largest $\Gamma_i$.
By Theorem~\ref{thm:main}(b) $|M_m-\log m|<\omega(m)$, and,
as was shown in the proof of Theorem~1(d), all
recurrent parts less than $\log m-\omega(m)$ are
asymptotically almost surely present in
$\Gamma$. Let
$$
\matrix{ \zeta_j:=|\{i:\Gamma_i=j\}|_{\vphantom{\bigm|}},\quad&
\lambda_j:=m(1-r)r^{j-1},\hfill \cr
k^-:=\km,\quad& k^+:=\kp.\cr}
$$
When $k^-\le k\le k^+$,
$$
\Pr(\zeta_j=k) ~\sim~ e^{-\lambda_j}\lambda_j^k/k!
$$
by the standard Poisson approximation for i.i.d.\ rare random
variables.
It should be well-known that $\{\zeta_j: k^-\le j\le k^+\}$
are asymptotically independent, but we include a proof since
we lack a reference.
For all fixed positive integers $m_1,\ldots, m_j$, we have
\begin{eqnarray*}
\Pr\left(\wedge_{k^-\le j\le k^+}\{\zeta_j=m_j\}\right)
&=&\frac{m! (\lambda_{k^-}/m)^{m_{k^-}}\cdots(\lambda_{k^+}/m)^{m_{k^+}}}
{(m_{k^-})!\cdots (m_{k^+})!(m-m_{k^-}+\cdots+m_{k^+})!}\\
& &\times
\left(1-\frac{\lambda_{k^-}+\ldots+\lambda_{k^+}}{m}\right)^{m-(m_{k^-}
+\ldots+m_{k^+})}\\
&\sim & \frac{\lambda_{k^-}^{m_{k^-}}\cdots
\lambda_{k^+}^{m_{k^+}}}{(m_{k^-})!\cdots
(m_{k^+})!}\exp(-(\lambda_{k^-}+\ldots +\lambda_{k^+})).
\end{eqnarray*}
Thus, with $k$ the largest part,
\begin{eqnarray}
\nonumber p_m &\sim& \sum_{k=k^-}^{k^+}
\Biggl(\prod_{j=k+1}^{k^+} e^{-\lambda_j}\Biggr)
\Biggl(\prod_{j=k^-}^k (1-e^{-\lambda_j})\Biggr)\\
&\sim& \label{eq:pm new}
\sum_{k=k^-}^{k^+} \exp\left(-mr^k\right)
\prod_{j=k^-}^k\left(1-\exp\left(-m(1-r)r^{j-1}\right)\right).
\end{eqnarray}
Equation~(\ref{eq:pm new}) is the same as the sum of (\ref{eq:qn(k)})
if $m=Cn/(1-r)$.
However, (\ref{eq:pm new}) was derived under the assumption that $m$
is an integer.
We now treat (\ref{eq:pm new}) as a function of real variable $m$,
say $f(m)$, and show that $f'(m)=o(1)$ as $m\to\infty$.
It then follows that $f(x)\sim f(\lfloor x\rfloor)$ as $x\to\infty$
and we will be done.
Call the terms in the sum (\ref{eq:pm new}) $T_k(m)$.
We have
\begin{eqnarray*}
|T_k'(m)|
&<& \left|\frac{T_k'(m)}{T_k(m)}\right|
~=~ \left|\left(\ln(T_k(m)\right)'\right|\le r^k
+ \sum_{j=k^-}^k\frac{(1-r)r^{j-1}}{\exp\left(m(1-r)r^{j-1}\right)-1} \\
&<& r^k + \sum_{j=k^-}^k\frac{(1-r)r^{j-1}}{m(1-r)r^{j-1}}
\le r^{k^-} + \frac{k-k^-+1}{m}
~<~ \frac{\omega_1(m)}{m}
\end{eqnarray*}
for some $\omega_1(m)\to\infty$ much slower than $m$.
Since there are only $2\omega(m)$ values for $k$, $f'(m)=o(1)$.
The oscillation is associated with the imaginary poles, which are at
$2k\pi i/\ln(1/q)$ in the notation of~\cite{HK}.
When the result is translated back from $m$ to $n$, we obtain
the same period as $P$ in~(\ref{eq:oscillation}).
\medskip
We now prove (f).
It follows from Theorem~\ref{thm:Poisson} that
$$
g_n(k)~\sim\hskip-.05in
\sum_{j>\log(Cn)-\omega(n)}\hskip-.1in
\Pr(\zeta_{j}=k,\zeta_{j+1}=\zeta_{j+2}=\cdots=0).
$$
Setting $j=\ell+\lfloor\log(Cn)\rfloor$ and
$\delta(n)=Cn-\lfloor\log(Cn)\rfloor$,
$$
g_n(k)\sim
\sum_{\ell=-\infty}^{\infty}\frac{r^{k(\ell-\delta(n))}}{k!}
\prod_{i\ge\ell}\exp\left(-r^{i-\delta(n)}\right)
\sim
\sum_{\ell=-\infty}^{\infty}\frac{r^{k(\ell-\delta(n))}}{k!}
\exp\left(\frac{-r^{\ell-\delta(n)}}{1-r}\right).
$$
It follows from Poisson's summation formula \cite{Olver} that
$$
g_n(k) ~\sim~
\sum_{\ell=-\infty}^{\infty}\int_{-\infty}^{\infty}
\frac{1}{k!}\exp(-2\pi i\ell t)r^{k(t-\delta(n))}
\exp\left(\frac{-r^{t-\delta(n)}}{1-r}\right)dt.
$$
Setting $z=\frac{r^{t-\delta(n)}}{1-r}$,
\begin{eqnarray*}
g_n(k)&\sim&
\frac{(1-r)^k\log e}{k!}\sum_{\ell=-\infty}^{\infty}
\exp(-2\pi i\ell(\delta(n)-\log(1-r)))
\int_{0}^{\infty}e^{-z}z^{k-1+2\pi i\ell\log e}dz\\
&\sim & \frac{(1-r)^k\log e}{k!}\sum_{\ell=-\infty}^{\infty}
\Gamma\left(k+2\pi i\ell\log e\right)
\exp\left(-2\pi i\ell\log\frac{Cn}{1-r}\right)\\
&\sim &
\frac{(1-r)^k}{k!}P_k\left(\frac{Cn}{1-r}\right)+\frac{(1-r)^k\log
e}{k}.
\end{eqnarray*}
This completes the proof of (f).
\medskip
We now prove (g).
By Lemma~\ref{lemma:zeta sum} and Theorem~\ref{thm:main}(b)
we may limit our attention to parts $j$ for which
$|j-\log n|\le \omega(n)$.
By Theorem~\ref{thm:Poisson}, the probability that part $j$
appears with multiplicity $k$ is asymptotically
$e^{-\mu_j}\mu_j^k/k!$ where $\mu_j=Cnr^j$.
Using the Poisson summation formula as in the proof of (f),
the expected number of parts of multiplicity $k$ is asymptotic to
\begin{eqnarray*}
& &\frac{1}{k!}\sum_j \exp\left(-Cnr^j\right)(Cnr^j)^k\\
&\sim
&\frac{1}{k!}\sum_{\ell=-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\left(-2i\pi
\ell t-r^{t-\delta(n)}\right)r^{k(t-\delta(n))}dt\\
&\sim &\frac{\log e}{k!}\sum_{\ell=-\infty}^{\infty}\exp\left(-2i\pi
\ell \log(Cn)\right)\Gamma\left(k+2i\pi \ell\log e\right)\\
&\sim &\frac{P_k(Cn)}{k!}+\frac{\log e}{k}.
\end{eqnarray*}
The claim about $m_n(k)$ follows from the fact that
$m_n(k)=\Ex(D_n(k)/D_n)$ and the tight concentration of $D_n$ in (c)---an
argument used by Louchard~\cite{Lo} for unrestricted compositions.
\EndProof
\vskip 20pt
{\baselineskip 12pt \frenchspacing
\begin{thebibliography}{99}
\bibitem{AK}
M. Archibald and A. Knopfmacher,
The largest missing value in a composition of an integer,
{\it Discrete Math.} {\bf311} (2011) 723--731.
\bibitem{BC1}
E.A. Bender and E.R. Canfield,
Locally restricted compositions I.
Restricted adjacent differences,
{\it Elec. J. Combin.} {\bf 12}(1) (2005) R57, 27pp.
\bibitem{BC2}
E.A. Bender and E.R. Canfield,
Locally restricted compositions II.
General restrictions and infinite matrices,
{\it Elec. J. Combin.} {\bf 16} (2009) R108, 35pp.
\bibitem{BC3}
E.A. Bender and E.R. Canfield,
Locally restricted compositions III.
Adjacent-part periodic inequalities,
{\it Elec. J. Combin.} {\bf 17} (2010) R145 9pp.
\bibitem{BCG}
E.A. Bender, E.R. Canfield, and Z.C. Gao
Locally Restricted Compositions IV,
to appear in the proceedings of
The 23rd International Meeting on Probabilistic, Combinatorial and
Asymptotic Methods for the Analysis of Algorithms,
Montreal, Canada, June, 2012.
\bibitem{FGD}
P. Flajolet, X. Gourdon and P. Dumas,
Mellin transform and asymptotics: Harmonic sums,
{\it Theoret. Comput. Science} {\bf 144} (1995) 3--58.
\bibitem{FS} P. Flajolet and R. Sedgewick,
{\it Analytic Combinatorics}, Cambridge University Press, 2009.
\bibitem{GW} Z.C. Gao and N.C. Wormald,
The distribution of the maximum vertex degree in random planar maps,
{\it J. Combin. Theory, Ser. A} {\bf 89} (2000) 201--230.
\bibitem{GH}
W.M.Y. Goh and P. Hitczenko,
Average number of distinct part sizes in a random Carlitz composition,
{\it Europ. J. Combinat.} {\bf 23} (2002) 647--657.
\bibitem{HK}
P. Hitczenko and A. Knopfmacher,
Gap-free compositions and gap-free samples of geometric random variables,
{\it Discrete Math.} {\bf 294} (2005) 225--239.
\bibitem{HL}
P. Hitczenko and G. Louchard,
Distinctness of compositions of an integer: A probablistic analysis,
{\it Random Structures and Algorithms} {\bf 19} (2001) 407--437.
\bibitem{HRS}
P. Hitczenko, C. Rousseau and C.D. Savage,
A generating functionology approach to a problem of Wilf,
{\it J. Comput. and Appl. Math.} {\bf 142} (2002) 107--114.
\bibitem{HSv}
P. Hitczenko and C.D. Savage,
On the multiplicity of parts in a random composition of a large integer,
{\it SIAM J. Discrete Math.} {\bf 18} (2004) 418--435.
\bibitem{HS}
P. Hitczenko and G. Stengle,
Expected number of distinct part sizes in a random integer composition,
{\it Combinat. Probab. Comput.} {\bf 9} (2000) 519--527.
\bibitem{HY}
H.-K. Hwang and Y. Yeh,
Measures of distinctness for random partitions and compositions of an integer,
{\it Adv. Appl. Math.} {\bf 19} (1999) 378--414.
\bibitem{JVZ} G. Jakli\v c, V. Vitrih and E. \v Zagar,
Closed form formula for the number of restricted compositions,
{\it Bull. Aust. Math. Soc.} {\bf 81} (2010) 289--297.
\bibitem{Kato}
T. Kato, {\it Perturbation Theory for Linear Operators}, Springer (1980).
\bibitem{Khe}
B.L. Kheyfets, The number of part sizes of a given multiplicity in a random
Carlitz composition, {\em Adv. in Appl. Math.} {\bf 35} (2005) 335--354.
\bibitem{KP}
A. Knopfmacher and H. Prodinger,
On Carlitz compositions,
{\it Europ. J. Combinat.} {\bf 19} (1998) 579--589.
\bibitem{Lo}
G. Louchard,
The number of distinct part sizes of some multiplicity in compositions of an
integer. A probabilistic analysis,
{\it Discrete random walks} ({\it Paris}, 2003) 155--170 (electronic).
{\it Discrete Math. Theor. Comput. Sci. Proc., AC, Assoc. Discrete Math.
Theor. Comput. Sci., Nancy}, 2003.
\bibitem{LP}
G. Louchard and H. Prodinger,
Probabilistic analysis of Carlitz compositions,
{\it Discrete Math. and Theoret. Comput. Sci.} {\bf 5} (2002) 71--96.
\bibitem{OR}
A.M. Odlyzko and L.B. Richmond,
On the compositions of an integer,
in {\it Combinatorial mathematics, VII},
(Proc. Seventh Australian Conf., Univ. Newcastle, Newcastle, 1979), 199-.210.
\bibitem{Olver}
F.W.J. Olver,
{\it Asymptotics and Special Functions},
Academic Press, New York, 1974.
\end{thebibliography}}
\end{document}
**