\documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=4.00.0.2321} %TCIDATA{Created=Sunday, September 07, 2003 19:38:35} %TCIDATA{LastRevised=Sunday, September 07, 2003 19:50:49} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=40 LaTeX article.cst} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \begin{center} {\large \textbf{Moment Generating Functions for Adjacency Statistics}}% \footnote{% Based on work supported by the National Science Foundation under Grant No. 0097392.}\$0pt] Don Rawlings, Lawrence Sze, and Mark Tiefenbruck \\[0pt] Math Department, California Polytechnic State University \\[0pt] San Luis Obispo, Ca. 93407 \end{center} \noindent \hspace{.38in}\textbf{Key words} adjacency, variation, oscillation \section{Introduction} For integers m,n\geq 1, let [m]^{n} denote the set of sequences % w=x_{1}x_{2}\ldots x_{n} of length n with each x_{i}\in \{1,2,\ldots ,m\} . For a real-valued function f on [m]^{2}, the f-\emph{adjacency number} of w=x_{1}x_{2}\ldots x_{n}\in \lbrack m]^{n} is defined to be \[ \limfunc{adf}w=\sum_{k=1}^{n-1}f(x_{k}x_{k+1}).$% Some specializations of the $f$-adjacency number have been considered elsewhere. For instance, if $f(xy)$ is 1 when \$x