\documentclass[12pt,reqno]{article}
\usepackage[usenames]{color}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amscd}
\usepackage[colorlinks=true,
linkcolor=webgreen,
filecolor=webbrown,
citecolor=webgreen]{hyperref}
\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}
\usepackage{color}
\usepackage{fullpage}
\usepackage{float}
\usepackage{psfig}
\usepackage{graphics,amsmath,amssymb}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{latexsym}
\usepackage{epsf}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\evensidemargin}{.1in}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.9in}
\newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}}
\begin{document}
\begin{center}
\epsfxsize=4in
\leavevmode\epsffile{logo129.eps}
\end{center}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\begin{center}
\vskip 1cm{\LARGE\bf Generating Functions for Wilf Equivalence
\\
\vskip .1in
Under Generalized Factor Order
}
\vskip 1cm
\large
Thomas Langley \\
Department of Mathematics \\
Rose-Hulman Institute of Technology \\
Terre Haute, IN 47803 \\
USA\\
\href{mailto:langley@rose-hulman.edu}{\tt langley@rose-hulman.edu}\\
\ \\
Jeffrey Liese \\
Department of Mathematics \\
California Polytechnic State University \\
San Luis Obispo, CA 93407-0403 \\
USA\\
\href{mailto:jliese@calpoly.edu}{\tt jliese@calpoly.edu}\\
\ \\
Jeffrey Remmel \\
Department of Mathematics \\
University of California, San Diego \\
La Jolla, CA 92093-0112 \\
USA\\
\href{mailto:remmel@math.ucsd.edu}{\tt remmel@math.ucsd.edu}
\end{center}
\vskip .2 in
\begin{abstract}
Kitaev, Liese, Remmel, and Sagan recently defined generalized factor
order on words comprised of letters from a partially ordered set $(P,
\leq_P)$ by setting $u \leq_P w$ if there is a contiguous subword $v$
of $w$ of the same length as $u$ such that the $i$-th character of $v$
is greater than or equal to the $i$-th character of $u$ for all $i$.
This subword $v$ is called an embedding of $u$ into $w$. For the case
where $P$ is the positive integers with the usual ordering, they
defined the weight of a word $w = w_1\ldots w_n$ to be $\text{wt}(w) =
t^{n} x^{\sum_{i=1}^n w_i}$, and the corresponding weight generating
function $F(u;t,x) = \sum_{w \geq_P u} \text{wt}(w)$. They then
defined two words $u$ and $v$ to be Wilf equivalent, denoted $u
\backsim v$,
if and only if $F(u;t,x) = F(v;t,x)$. They also defined the related
generating function $S(u;t,x) = \sum_{w \in \mathcal{S}(u)}
\text{wt}(w)$ where $\mathcal{S}(u)$ is the set of all words $w$ such
that the only embedding of $u$ into $w$ is a suffix of $w$, and showed
that $u \backsim v$ if and only if $S(u;t,x) = S(v;t,x)$. We continue
this study by giving an explicit formula for $S(u;t,x)$ if $u$ factors
into a weakly increasing word followed by a weakly decreasing word. We
use this formula as an aid to classify Wilf equivalence for all words
of length 3. We also show that coefficients of related generating
functions are well-known sequences in several special cases. Finally,
we discuss a conjecture that if $u \backsim v$ then $u$ and $v$ must be
rearrangements, and the stronger conjecture that there also must be a
weight-preserving bijection $f$ on words over the positive integers
such that $f(w)$ is a rearrangement of $w$ for all $w$, and $w$
embeds $u$ if and only if $f(w)$ embeds $v$.
\end{abstract}
\long\def\symbolfootnote[#1]#2{\begingroup
\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup}
\newcommand{\la}{\lambda}
\newcommand{\La}{\Lambda}
\newcommand{\sg}{\sigma}
\newcommand{\Ta}{\Theta}
\def\mn{{\mbox{-}}}
\newcommand\mpg[1]{\marginpar{\raggedleft\tiny#1}}
\def\W{\mathcal{W}}
\def\S{\mathcal{S}}
\def\A{\mathcal{A}}
\def\B{\mathcal{B}}
\def\C{\mathcal{C}}
\def\D{\mathcal{D}}
\def\Z{\mathbb{Z}}
\def\F{\mathcal{F}}
\def\P{\mathbb{P}}
\def\mod{\text{ mod }}
% newcommands to be used in regular mode
\newcommand{\fig}[3]
{\begin{figure}[ht] \centerline{\scalebox{#1}{\epsfig{file=#2.eps}}}
\vspace{-1mm} \caption{#3} \label{fig:#2}
\end{figure}}
\newcommand{\eref}[1]{(\ref{equation:#1})}
\newcommand{\sref}[1]{Section \ref{sub:#1}}
\newcommand{\tref}[1]{Theorem \ref{theorem:#1}}
\newcommand{\lref}[1]{Lemma \ref{lemma:#1}}
\newcommand{\fref}[1]{Figure \ref{figure:#1}}
\section{Introduction and definitions} \label{sec:intro}
Kitaev, Liese, Remmel, and Sagan \cite{KLRS} recently introduced the
generalized factor order on words comprised of letters from a partially ordered set (poset). That is,
let $\mathcal{P} =(P, \leq_P)$ be a poset and let $P^*$ be the Kleene closure of $P$ so that
$$
P^* = \{w = w_1 w_2 \ldots w_n \: | \: n \geq 0 \text{ and } w_i \in P \text{ for all } i \}.
$$
For $w \in P^*$, let $|w|$ denote
the number of characters in $w$. Then for any $u, w \in P^*$, $u$ is less than or equal to $w$ in the
\textit{generalized factor order} relative to $\mathcal{P}$, written
$u \leq_{\mathcal{P}} w$, if
there is a string $v$ of $|u|$ consecutive characters in $w$ such that
the $i$-th character of $v$ is greater than or equal to the $i$-th
character of
$u$ under $\leq_\mathcal{P}$ for each $i$, $1 \leq i \leq |u|$. If $u \leq_{\mathcal{P}} w$, we will also say that $w$ \textit{embeds} $u$, and that $v$ is an \textit{embedding of} $u$ \textit{into} $w$.
We will primarily be interested in the poset $\mathcal{P}_1 = (\P, \leq)$, where $\P$ is the set of positive integers and
$\leq$ is the usual total order on $\P$. In this case, for example, $u = 321 \leq_{\mathcal{P}_1} w = 142322$,
and 423 and 322 are embeddings of $u$ into $w$. Kitaev, Liese, Remmel, and Sagan \cite{KLRS}
noted that generalized factor order is related to
generalized subword order, in which the characters of $v$ are not required to be adjacent \cite{SV}.
Kitaev, Liese, Remmel, and Sagan \cite{KLRS} defined Wilf equivalence under
the generalized factor order on the positive integers in the following way. For
$w = w_1 \ldots w_n \in \P^*$, let
$\Sigma(w) = \sum_{i=1}^n w_i$ and define the \emph{weight} of $w$ to be
$\text{wt}(w) = t^{n} x^{\Sigma(w)}$. Then define
$$
\F(u) = \{w \in \P^* \: | \: u \leq_{\mathcal{P}_1} w \},
$$
and the related generating function
$$
F(u;t,x) =
\sum_{w \in \F(u)} \text{wt}(w).
$$
Two words $u, v \in \P^*$ are then said to
be \emph{Wilf equivalent}, denoted $u \backsim v$,
if and only if
$F(u;t,x) = F(v;t,x)$. Kitaev, Liese, Remmel, and Sagan \cite{KLRS} noted that this idea, while inspired by
the notion of Wilf equivalence used in the theory of pattern avoidance, is different, since the
partial order in question is not that of pattern containment. More information about Wilf
equivalence in the pattern avoidance context is contained in the survey article by Wilf \cite{Wilf}.
In proving results about Wilf equivalence, it is often convenient to study the sets
\begin{eqnarray*}
\S(u) & = & \{w \in \P^* \: | \: u \leq_{\mathcal{P}_1} w \text{ and the last } |u| \text{ characters of } w \text{ form the only } \\
&& \hspace{.62in} \text{ embedding of } u \text{ into } w \}, \\
\W(u) & = & \{w \in \P^* \: | \: u \leq_{\mathcal{P}_1} w \text{ and } |w|=|u| \},\text{ and}\\
\A(u) & = & \{w \in \P^* \: | \: u \not \leq_{\mathcal{P}_1} w \}
\end{eqnarray*}
and the corresponding weight generating functions
\begin{eqnarray*}\label{basicgf}
S(u;t,x) &=& \sum_{w \in \S(u)} \text{wt}(w), \\
W(u;t,x) &=& \sum_{w \in \W(u)} \text{wt}(w), \text{ and } \\
A(u;t,x) &=& \sum_{w \in \A(u)} \text{wt}(w).
\end{eqnarray*}
Kitaev, Liese, Remmel, and Sagan \cite{KLRS} proved that $F(u;t,x)$, $S(u;t,x)$,
and $A(u;t,x)$ are rational. They constructed a non-deterministic finite
automaton for each $u \in \P^*$ that recognizes $\S(u)$,
implying that $S(u;t,x)$ is rational. That the others are rational follows
from the fact that the weight generating function for all words in $\P^*$ is
\begin{eqnarray*}
\sum_{w \in \P^*} \text{wt}(w) & = & \frac{1}{1-\sum_{n \geq 1} tx^n} \\
& = & \frac{1}{1 - tx/(1-x) } \\
& = & \frac{1-x}{1-x-tx},
\end{eqnarray*}
and therefore
\begin{equation}\label{eq:FS}
F(u;t,x) = S(u;t,x) \frac{1-x}{1-x - tx}
\end{equation}
and
$$
F(u;t,x) = \frac{1-x}{1-x-tx} - A(u;t,x).
$$
We also note that $W(u;t,x)$ is
rational since
$$
W(u;t,x) = \frac{t^{|u|}x^{\Sigma(u)}}{(1-x)^{|u|}}.
$$
From (\ref{eq:FS}), we have that $F(u;t,x) = F(v;t,x)$ if and only if $S(u;t,x) = S(v;t,x)$,
and therefore $u \backsim v$ if and only if $S(u;t,x) = S(v;t,x)$. Much of our
work will be centered around computing explicit formulas for $S(u;t,x)$ for certain
words $u$. In particular,
Kitaev, Liese, Remmel and Sagan \cite{KLRS} gave two examples
of classes of words $u$ such that $S(u;t,x)$ has a
simple form. That is,
they proved that if $u = 1~2~3 \ldots n-1~n$ or $u =1^kb^\ell$ for
some $k \geq 0$, $\ell \geq 1$, and $b \geq 2$, then
$S(u;t,x) = \frac{x^s t^r}{P(u;t,x)}$ for some polynomial $P(u;t,x)$, and
produced an explicit expression for $P(u;t,x)$ in each case.
We shall show that there is a much richer class of
of words $u$ such that $S(u;t,x)$ has this same form. Specifically,
for any word $u$, let
$u_{inc}$ be the longest weakly increasing prefix of $u$. If
$u = u_{inc}v$ and $v$ is weakly decreasing, then we shall
say that $u$ has an \textit{increasing/decreasing factorization} and
denote $v$ as $u_{dec}$. Thus if $u = u_1 u_2 \ldots u_n$
has an increasing/decreasing factorization, then either
$u_1 \leq \cdots \leq u_n$, in which case $u_{inc}= u$ and
$u_{dec}$ is the empty string $\varepsilon$, or
there is a $k u_{k+1} \geq \cdots \geq
u_n$, in which case $u_{inc} = u_1 \ldots u_k$ and $u_{dec} =
u_{k+1} \ldots u_n$. For the theorem that follows, we define
$$
D^{(i)}(u) = \{n-i+j: 1 \leq j \leq i \text{ and } u_j > u_{n-i+j}\}
$$
and $d_i(u) = \sum_{n-i+j \in D^{(i)}(u)} (u_j - u_{n-i+j})$.
For example, if $u = 1~2~3~4~4~3~1~1$ and $i=5$, then
by considering the diagram
$$
\begin{array}{cccccccc}
1 & 2 & 3 & 4 & 4 & 3 & 1 & 1 \\
& & & 1 & 2 & 3 & \underline{4} & \underline{4} \end{array}
$$
we see that
$D^{(5)}(u) = \{7,8\}$ and $d_5(u) = (4-1) + (4-1) = 6$.
One of our main results is the following theorem.
%___________________________________
\begin{theorem}\label{thm:S}
Let $u=u_1 u_2 \ldots u_n \in \P^*$ have an increasing/decreasing factorization. For $1 \leq i \leq n-1$, let
$s_i = u_{i+1} u_{i+2} \ldots u_n$ and $d_i = d_i(u)$. Also let $s_{n} = \varepsilon$ and
$d_n = 0$.
Then
$$
S(u;t,x) = \frac{t^{n} x^{\Sigma(u)}}{t^{n} x^{\Sigma{(u)}}+(1-x-tx)\sum_{i=1}^{n} t^{n-i} x^{d_i+\Sigma(s_i)}(1-x)^{i-1}}.
$$
\end{theorem}
%______________________________________
Since the words $u = 1~2~3 \ldots n-1~n$ or $u =1^kb^\ell$ for
some $k \geq 0$, $\ell \geq 1$, and $b \geq 2$ clearly have increasing/decreasing factorizations,
Theorem \ref{thm:S} covers both of the cases
proved by Kitaev, Liese, Remmel and Sagan \cite{KLRS}.
Theorem~\ref{thm:S} will lead us to the other main results in our work. First,
we will use Theorem \ref{thm:S}, as well as a slight modification in a special case, to
completely
classify the Wilf equivalence classes of $\mathcal{P}_1$ for all
words of length 3. We will also compute $S(u;t,x)$, along with $F(u;t,x)$ and $A(u;t,x)$,
for some simple words and show that the coefficients in these generating functions
are often well-known sequences.
Next,
Theorem \ref{thm:S} will allow us
to show that if $u$ and $v$ are words with
increasing/decreasing factorizations, then $u \backsim v$ if and only
if $v$ is a rearrangement of the letters of $u$. This shows
that words with increasing/decreasing factorizations satisfy
the following conjecture of Kitaev, Liese, Remmel, and Sagan \cite{KLRS}.
\begin{conjecture}[Kitaev, Liese, Remmel, Sagan]\label{con:weak}
If $u \backsim v$, then $v$ is a rearrangement of $u$.
\end{conjecture}
We shall call this conjecture the {\em weak rearrangement conjecture}.
In fact, we conjecture something much stronger is true.
\begin{conjecture} \label{con:strong}
If $u \backsim v$, then there is a weight preserving bijection
$f:\P^* \rightarrow \P^*$ such that for all $w \in \P^*$,
$f(w)$ is a rearrangement of $w$ and
$w \in \F(u) \iff f(w) \in \F(v)$.
\end{conjecture}
We will call such a bijection $f$
a \emph{rearrangement map} that \emph{witnesses} $u \backsim v$ and
refer to this conjecture as the {\em strong rearrangement conjecture}. All the Wilf equivalences proved by Kitaev, Liese, Remmel, and Sagan in Section 4 of \cite{KLRS}
were proved by a constructing a rearrangement map that witnessed the given
Wilf equivalence.
We investigate the rearrangement conjectures by considering the class of finite posets $\mathcal{P}_{[m]} = ([m],\leq)$, where $[m] = \{1, \ldots, m\}$ and
$\leq$ is the usual total order on $\mathbb{P}$.
For any word $w \in [m]^*$ and $i \in [m]$, let $c_i(w)$ equal the number of occurrences of $i$ in $w$.
Then we introduce variables $x_1, x_2, \ldots, x_m$, and define the weight of $w$, $W_{[m]}(w)$, to be $W_{[m]}(w) = \prod_{i=1}^m x_i^{c_i(w)}$. To define Wilf equivalence
in this context, we set
\begin{eqnarray*}
F(u; x_1, \ldots , x_m) & = & \sum_{w \in \F(u) \cap [m]^*} W_{[m]}(w),
\end{eqnarray*}
and define $u, v \in [m]^*$ to be Wilf equivalent with respect to the poset $\mathcal{P}_{[m]}$, denoted $u \backsim_{[m]} v$, if and only $F(u; x_1, \ldots , x_m) = F(v; x_1, \ldots , x_m)$.
We will also have use for the related generating functions
\begin{eqnarray*}
W(u; x_1, \ldots , x_m) & = & \sum_{w \in \W(u) \cap [m]^*} W_{[m]}(w), \\
S(u; x_1, \ldots , x_m) & = & \sum_{w \in \S(u)\cap [m]^*} W_{[m]}(w), \ \mbox{and} \\
A(u; x_1, \ldots , x_m) & = & \sum_{w \in \A(u)\cap [m]^*} W_{[m]}(w).
\end{eqnarray*}
Note that we have dropped the $t$ dependence in these generating functions
since length is recorded by the
number of variables in a monomial. We now have
$$
\sum_{w \in [m]^*} W_{[m]}(w) = \frac{1}{1 - \sum_{i=1}^m x_i}.
$$
Thus since $\F(u)\cap [m]^* = (\S(u)\cap [m]^*)[m]^*$ and
$A(u) \cap [m]^* = [m]^* - (\F(u) \cap [m]^*)$, we have that
\begin{eqnarray*}
F(u; x_1, \ldots , x_m) &=& S(u; x_1, \ldots , x_m)\frac{1}{1 - \sum_{i=1}^m x_i} \ \mbox{and} \\
A(u; x_1, \ldots , x_m) &=& \frac{1}{1 - \sum_{i=1}^m x_i} - F(u; x_1, \ldots , x_m)
\end{eqnarray*}
so that if any one of $F(u; x_1, \ldots , x_m)$,
$A(u; x_1, \ldots , x_m)$, or $S(u; x_1, \ldots , x_m)$ is rational,
then so are the other two. It follows from Theorem 8.2 of
\cite{KLRS} that $S(u; x_1, \ldots , x_m)$ is rational for
all $m \geq 1$, so $F(u; x_1, \ldots , x_m)$ and
$A(u; x_1, \ldots , x_m)$
are also rational for all $m \geq 1$. Also note that if
$u =u_1 \ldots u_n$, then
$$
W(u;x_1, \ldots ,x_m) = \prod_{r=1}^n \sum_{s=u_r}^m x_s,
$$
so $W(u;x_1, \ldots ,x_m)$ is rational. We will show
that if $u \backsim_{[m]} v $ for some $m$, then there is a
rearrangement map that witnesses the equivalence $u \backsim v$. This gives us
a way to test the strong rearrangement conjecture for any
particular pair of words $u,v \in \P^*$. We will also give an analogue
of Theorem~\ref{thm:S} for these posets.
The outline of this paper is as follows. In Section~\ref{sec:S},
we prove Theorem \ref{thm:S} and show that the weak rearrangement conjecture
holds for words with increasing/decreasing factorizations. In Section~\ref{sec:sequences},
we
compute $F(u;t,x)$, $S(u;t,x)$ and $A(u;t,x)$ for some simple words. In particular,
we show that the sequences of coefficients that arise in the expansions around $x = 0$ of
$F(k;1,x)$, $S(k;1,x)$, $A(k;1,x)$, $S(1k1;1,x)$ and $A(1k1;1,x)$ as $k$ varies
have appeared in
the On-line Encyclopedia of Integer Sequences (OEIS).
We
follow this with the classification of the Wilf equivalence classes
of words of length 3 in Section~\ref{sec:3}. The results
of Sections \ref{sec:S} and \ref{sec:3} allow us to compute
$S(\sigma;t,x)$ and $A(\sigma;t,x)$ for all permutations
in the symmetric group $S_3$ as there are only two Wilf equivalences
classes for such permutations. In these cases,
the coefficients that arise in
the expansions of $S(\sigma;1,x)$ and $A(\sigma;1,x)$
around $x=0$ do not correspond to any sequences that have appeared
in the OEIS. We discuss the strong rearrangement conjecture in
Section~\ref{sec:rearrangement},
as well as the analogue of Theorem~\ref{thm:S} for the posets $\mathcal{P}_{[m]}$.
We conclude with a few remarks about further work in Section~\ref{sec:conclusion}.
%%%%%%%%%%%
\section{Words such that $S(u;t,x) = \frac{x^st^r}{P(u;t,x)}$ where $P(u;t,x)$
is a polynomial.} \label{sec:S}
In this section we prove Theorem~\ref{thm:S}, and show that Conjecture~\ref{con:weak} holds for words with an increasing/decreasing factorization. \\
\noindent \textit{Proof of Theorem~\ref{thm:S}.}
Let $u=u_1 u_2 \ldots u_n \in \P^*$ have an increasing/decreasing factorization. If $w = w_1 \ldots w_m \in \mathcal{S}(u)$, then $w_1 \ldots w_{m-n} \in \A(u)$ and $u \leq
w_{m-n+1} \ldots w_{m}$. However if
$v \in \A(u)$ and $z = z_1 \ldots z_n$ is such that
$u \leq z$, then it may not be the case that
$w = vz \in \S(u)$ because there might be another
embedding of $u$ in the last $2n-1$
letters of $w$, starting in $v$ and ending in $z$.
Of course, there can be
no embedding of $u$ which starts to the left of the last $2n-1$ letters of
$w$ since $v \in \A(u)$.
For each $1 \leq i \leq n-1$,
we define $\S^{(i)}(u)$ to be set of all words $w = w_1 \ldots w_m$ such
that
\begin{description}
\item[(i)] $u \leq w_{m-n+1} \ldots w_m$ (so that $u$ embeds into the suffix
of length $n$ of $w$)
and
\item[(ii)] the left-most embedding of $u$ into $w$ starts at position
$m-2n+i+1$.
\end{description}
Note that $\S^{(i)}(u)$ is empty when $m-2n+i+1$ is non-positive. We then let
$$
S^{(i)}(u;t,x) = \sum_{w \in \S^{(i)}(u)} \text{wt}(w) = \sum_{w \in \S^{(i)}(u)} x^{\Sigma(w)} t^{|w|}.
$$
Thus
\begin{equation}\label{eq:S2}
\S(u) = \A(u)\mathcal{W}(u)- \bigcup_{i=1}^{n-1} \S^{(i)}(u).
\end{equation}
Now,
\begin{eqnarray}\label{eq:S3}
\sum_{w \in \A(u)\mathcal{W}(u)} x^{\Sigma(w)} t^{|w|} &=&
A(u;t,x)\frac{t^nx^{\Sigma(u)}}{(1-x)^n} \nonumber \\
&=& \frac{(1-x)}{(1-x-tx)}(1-S(u;t,x))\frac{t^nx^{\Sigma(u)}}{(1-x)^n}.
\end{eqnarray}
We claim that we have the following lemma.
%______________________________________
\begin{lemma}\label{lem:S}
Let $u=u_1 u_2 \ldots u_n \in \P^*$ have an increasing/decreasing factorization, and define
$d_i$ and $s_i$ as in Theorem~\ref{thm:S}. Then
for $1 \leq i \leq n-1$,
$$
S^{(i)}(u;t,x) = S(u;t,x) t^{n-i}x^{d_i+\Sigma(s_i)} \left(\frac{1}{1-x}\right)^{n-i}.
$$
\end{lemma}
%_______________________________________
Given Lemma \ref{lem:S}, it is easy to complete the
proof of Theorem \ref{thm:S}. That is, our definitions ensure that
$\S^{(1)}(u), \S^{(2)}(u), \ldots,
\S^{(n-1)}(u)$ are pairwise disjoint, so that
\begin{eqnarray*}
\sum_{w \in \bigcup_{i=1}^{n-1} \S^{(i)}(u)} x^{\Sigma(w)} t^{|w|} &=&
\sum_{i=1}^{n-1} S^{(i)}(u;t,x) \\
&=& S(u;t,x) \sum_{i=1}^{n-1} t^{n-i}x^{d_i+\Sigma(s_i)} \left(\frac{1}{1-x}\right)^{n-i}.
\end{eqnarray*}
Thus it follows from (\ref{eq:S2}) and (\ref{eq:S3}) that
\begin{eqnarray*}
S(u;t,x) & = & \frac{(1-x)}{(1-x-tx)}(1-S(u;t,x))\frac{t^nx^{\Sigma(u)}}{(1-x)^n}\\
&&- S(u;t,x) \sum_{i=1}^{n-1} \frac{x^{d_i+\Sigma(s)_i}t^{n-i}}{(1-x)^{n-i}}.
\end{eqnarray*}
Solving for $S(u;t,x)$ will yield the result in the theorem.
Thus we need only prove Lemma \ref{lem:S}. To this end, fix $i$,
$1 \leq i \leq n-1$, and suppose that $w =w_1 \ldots w_m \in \S^{(i)}(u)$.
If $\bar{w}= w_1 \ldots w_{m-n+i}$, then our definitions ensure
that
\begin{enumerate}
\item $\bar{w} \in \S(u)$,
\item $u_1 \ldots u_i \leq w_{m-n+1} \ldots w_{m-n+i}$ and
\item $s_i = u_{i+1} \ldots u_n \leq w_{m-n+i+1} \ldots w_m$.
\end{enumerate}
Now, the generating function of all words $v$ of length $n-i$ such
that $s_i \leq v$ is $\frac{x^{\Sigma (s_i)} t^{n-i}}{(1-x)^{n-i}}$.
So let $\bar{\S}^{(i)}(u)$ denote the set of all words
$\bar{w}$ that satisfy conditions 1 and 2, and let
$$\bar{S}^{(i)}(u;t,x) = \sum_{\bar{w} \in \bar{\S}^{(i)}(u)} x^{\Sigma(\bar{w})}t^{|\bar{w}|}.$$
Then
$$
S^{(i)}(u;t,x) =\bar{S}^{(i)}(u;t,x)\frac{x^{\Sigma (s_i)} t^{n-i}}{(1-x)^{n-i}}.
$$
Thus we need only show that
\begin{equation}\label{eq:S6}
\bar{S}^{(i)}(u;t,x) = x^{d_i} S(u;t,x).
\end{equation}
Now suppose that $v = v_1 \ldots v_p \in \bar{\S}^{(i)}(u)$. Then
let $\tilde{v} = \tilde{v}_1 \ldots \tilde{v_p}$ be the
word that results from $v$ by decrementing $v_{p-i+j}$ by $u_j - u_{n-i+j}$ if $n-i+j \in D^{(i)}(u)$ and leaving all other letters the same. If
$n-i+j \in D^{(i)}(u)$, then $v_{p-i+j} \geq u_j$, and hence
$\tilde{v}_{p-i+j} \geq u_{n-i+j}$. Thus it will still be the case that
$u$ embeds in the final segment of $\tilde{v}$ of length $n$ so that
$\tilde{v} \in \S(u)$. Thus to complete the proof of (\ref{eq:S6}),
we need only show that if we start with a word
$\tilde{v} = \tilde{v}_1 \ldots \tilde{v_p}$ in $\S(u)$ and
create a new word $v =v_1 \ldots v_p$ by incrementing $\tilde{v}_{p-i+j}$ by $u_j - u_{n-i+j}$ if $n-i+j \in D^{(i)}(u)$ and leaving all other letters the same, then $v \in \bar{\S}^{(i)}(u)$. Clearly $v$ satisfies condition (2) above.
The only question is whether $v$ is still in $\S(u)$. That is,
since we have incremented some letters in $\tilde{v}$ to get $v$, we
might have created a
new embedding of $u$ which starts to the left of position
$p-n+1$. If so, any such embedding must contain
at least one position of the form $p-i+j$ where $n-i+j \in D^{(i)}(u)$. However if $u_r$ is the letter in this new embedding of $u$ into $v$
which corresponds to position
$p-i+j$, then $r$ must be strictly greater than $n-i+j$.
But if $u= u_1 \leq \cdots \leq u_k > u_{k+1} \geq \cdots \geq u_n$,
then it must always be the case that $D^{(i)}(u) \subseteq \{k+1, \ldots, n\}$. That is, if $j < n-i+j \leq k$, then $u_j \leq u_{n-i+j}$ and
hence $n-i+j \not \in D^{(i)}(u)$.
Hence $u_{n-i+j} \geq u_r$.
But then $\tilde{v}_{p-i+j} \geq u_{n-i+j} \geq u_r$ which would mean
that there would have been an embedding of $u$ into $\tilde{v}$ which
started to the left of $p-n+1$. Since $\tilde{v}$ was assumed to
be in $\S(u)$, there can be no such embedding and hence
$v \in \S(u)$. Thus (\ref{eq:S6}) holds and the lemma is proved. \qed \\
To illustrate the ideas in the proof, consider $u = 1\:2\:6\:5\:3\:2$,
so that $u_{inc} = 1\: 2\: 6$ and $u_{dec} = 5\:3\:2$, and let $i=5$. Then elements of $\bar{\S}^{(5)}(u)$ must end in an embedding of $u$ in
the final six characters and an embedding of $1\:2\:6\:5\:3$ in the final five characters, as shown:
$$
\begin{array}{ccccccccc}
\tilde{v}= & \cdots & \bullet & \bullet & \bullet & \star & \star & \star \\
& & 1& 2 & 6 & 5 & 3 & 2 \\
& & & 1& 2 & 6 & 5 & 3 \\
\end{array},
$$
where the stars indicate the positions in $\tilde{v}$ that must be increased to form $v$.
Note that the stars all embed characters of $u_{dec}$, and that $d_5 = (6-5)+(5-3)+(3-2) =4$.
If $v$ were to contain a new embedding of $u$ to the left of the first original embedding, that
new embedding must end in the second or third position from the end:
$$
\begin{array}{cccccccccc}
v= & \cdots & \bullet & \bullet & \bullet & \bullet & \star & \star & \star \\
& & & 1& 2 & 6 & 5 & 3 & 2 \\
& & 1& 2 & 6 & 5 & 3 & 2 \\
\end{array}
$$
or
$$
\begin{array}{cccccccccccc}
v= & \cdots & \bullet & \bullet & \bullet & \bullet & \bullet & \star & \star & \star \\
& & & & 1& 2 & 6 & 5 & 3 & 2 & \\
& & 1& 2 & 6 & 5 & 3 & 2 & & & \\
\end{array}.
$$
But in both cases, the characters below the stars
are decreasing, so such an embedding would have already existed
in $\tilde{v}$.
It is worth noting here that the condition that $u$ has an increasing/decreasing
factorization is necessary for the technique in the proof of Lemma~\ref{lem:S}
to be valid. That is, if $u$ does not have an increasing/decreasing factorization,
there is always at least one index $i$ where words counted by
$\bar{S}^{(i)}(u;t,x)$ can not be formed by simply starting
with a word $\tilde{v} \in \S(u)$ and creating a
word $v =v_1 \ldots v_p$ by incrementing $\tilde{v}_{p-i+j}$ by $u_j - u_{n-i+j}$ if $n-i+j \in D^{(i)}(u)$ and leaving all other letters the same.
For example, consider $u=2112$ with $i=2$. Then $D^{(2)}(u) = \{3\}$ and
$d_2(u) =1$. However if we start with $\tilde{v} =
122112 \in \S(u)$ and increment $\tilde{v}_{5}$ to obtain $v$,
then $v=122122$ which is not in $\S(u)$ because there is an embedding
of $u$ which starts at position 2.
The problem here is that the second 1 in $u$ is followed by a larger character, and
also has a larger character to its left. A similar situation will always occur for
at least one $i$ when $u$ does not have an increasing/decreasing factorization. Experimental evidence suggests the following conjecture.
\begin{conjecture}
For $u \in \P^*$, $S(u; t,x)= \frac{x^s t^r}{P(u;t,x)}$ where
$P(u;t,x)$ is a polynomial if and only if $u$ has an increasing/decreasing factorization.
\end{conjecture}
It is a consequence of Corollary 4.2 in \cite{KLRS} that if
$u$ and $v$ have increasing/decreasing factorizations and
$u$ is a rearrangement of $v$, then $u \backsim v$.
We shall give a new proof of that fact here, as well as prove the
converse. That is, if $u$ and $v$ both have increasing/decreasing factorizations and $u \backsim v$, then $u$ and $v$ are rearrangements, showing that
the weak rearrangement conjecture holds for words with increasing/decreasing factorizations.
We begin with the following lemma.
%________________________________
\begin{lemma}\label{lem:rearrangements}
Suppose $u = u_1 \ldots u_n$ is a rearrangement of $v=v_1 \ldots v_n$ and
that $u$ and $v$ have increasing/decreasing factorizations.
For each $i$, $1 \leq i \leq n-1$, let
$s_i(u) = u_{i+1} \ldots u_n$, $s_i(v) = v_{i+1} \ldots v_n$,
$d_i(u) = \sum_{n-i+j \in D^{(i)}(u)} (u_j - u_{n-i+j})$, and
$d_i(v) = \sum_{n-i+j \in D^{(i)}(v)} (v_j - v_{n-i+j})$. Then
for all $1 \leq i \leq n-1$,
$$
d_i(u)+\Sigma(s_i(u)) = d_i(v)+\Sigma(s_i(v)).
$$
\end{lemma}
%______________________________
\begin{proof}
First suppose that $u = u_1 \ldots u_n$ where $u_1 \leq \cdots \leq u_n$.
Then for each $i$, $1 \leq i \leq n-1$, $d_i(u) =0$ and $\Sigma(s_i(u)) =
\sum_{j=i+1}^n u_j$. So it suffices to show that
$d_i(v)+\Sigma(s_i(v)) = \Sigma(s_i(u))$ for all $1 \leq i \leq n-1$ whenever
$v$ has an increasing/decreasing factorization and $v$ is a rearrangement
of $u$. So fix $i$, $1 \leq i \leq n-1$, and let $\sg = \sg_1 \ldots \sg_n$ be a permutation of $\{1,\ldots, n\}$ such that
$$v =
u_{\sg_1} \leq \cdots \leq u_{\sg_j} > u_{\sg_{j+1}} \geq \cdots \geq u_{\sg_n}.$$
Then let
\begin{eqnarray*}
A^i &=& \{s: s \leq i \text{ and } u_s \in \{u_{\sg_1}, \ldots ,u_{\sg_j}\}\}, \\
B^i &=& \{s: s > i \text{ and } u_s \in \{u_{\sg_1}, \ldots ,u_{\sg_j}\}\}, \\
C^i &=& \{s: s > i \text{ and } u_s \in \{u_{\sg_{j+1}}, \ldots ,u_{\sg_n}\}\},
\ \mbox{and} \\
D^i &=& \{s: s \leq i \text{ and } u_s \in \{u_{\sg_{j+1}}, \ldots ,u_{\sg_n}\}\}.
\end{eqnarray*}
For example, suppose $u =1~2~3~3~4~5~5~6~7~7$ and
$\sg = 2~3~4~9~10~8~7~6~5~1$ so that
$$
\begin{array}{ccccccccccccc}
v & = & u_2 & u_3 & u_4 & u_9 & u_{10} & u_8 & u_7 & u_6 & u_5 & u_1\\
& = & 2 & 3 & 3 & 7 & 7 & 6 & 5 & 5 & 4 & 1
\end{array}
$$
and $j=5$. Then for $i=6$, $A^6 =\{2,3,4\}$, $B^6 =\{9,10\}$, $C^6 = \{7,8\}$, and
$D^6 = \{1,5,6\}$. Let $a_i = |A^i|$, $b_i = |B^i|$, $c_i = |C^i|$, and
$d_i = |D^i|$. Then our definitions force $a_i+d_i =i$, $b_i+c_i = n-i$,
$a_i + b_i =j$, and $c_i +d_i =n-j$. For any set
$D = \{d_1 < \cdots < d_r\} \subseteq \{1, \ldots ,n\}$, let
\begin{eqnarray*}
D(u)\!\!\uparrow &=& u_{d_1}u_{d_2} \ldots u_{d_r} \ \mbox{and} \\
D(u)\!\!\downarrow &=& u_{d_r}u_{d_{r-1}} \ldots u_{d_1}.
\end{eqnarray*}
Thus $v = A^i(u)\!\!\uparrow B^i(u)\!\!\uparrow C^i(u)\!\!\downarrow D^i(u)\!\!\downarrow$ and
$\Sigma(B^i(u)\!\!\uparrow) + \Sigma(C^i(u)\!\!\downarrow) =\Sigma(s_i(u))$.
We then have four cases to consider depending on whether
$v_i \in A^i(u)\!\!\uparrow$, $v_i \in B^i(u)\!\!\uparrow$, $v_i \in C^i(u)\!\!\downarrow$, or $v_i \in D^i(u)\!\!\downarrow$. \\
\ \\
{\bf Case 1.} $v_i \in A^i(u)\!\!\uparrow$.\\
\ \\
In this case, it must be that $i = a_i$ and $A^i(u)\!\!\uparrow =u_1 \ldots u_i$.
But then $D^i = \emptyset$ and $s_i(v) = B^i(u)\!\!\uparrow C^i(u)\!\!\downarrow$,
a rearrangement of $s_i(u)$. Moreover, it will
be the case that $v_j \leq v_{n-i+j}$ for $j=1, \ldots, i$ so that
$d_i(v) = 0$. Thus $d_i(v) + \Sigma(s_i(v)) = \Sigma(s_i(u))$ as
desired. As an example, with $u$ as in the previous example, consider
$$
\begin{array}{cccccccc|cc|cc}
v & = & u_1 & u_2 & u_3 & u_4 & u_{5} & u_6 & u_9 & u_{10} & u_8 & u_7\\
& = & 1 & 2 & 3 & 3 & 4 & 5 & 7 & 7 & 6 & 5 \end{array}
$$
so that $j=8$, and again let $i = 6$. Then as indicated by the dividers, $A^6(u)\!\!\uparrow= u_1 \ldots u_6$ so that $a_i = i = 6$, $B^6(u)\!\!\uparrow = u_9 u_{10}$ and
$C^6(u)\!\!\downarrow = u_8 u_7$.\\
\ \\
{\bf Case 2.} $v_i \in B^i(u)\!\!\uparrow$.\\
\ \\
In this case $a_i * i\}$. Thus any letter in
$B_1^i(u)$ is greater than or equal to every letter in $D^i(u)\!\!\downarrow$
so that such letters will contribute
$\Sigma(B^i_1(u)) - \Sigma(D^i(u)\!\!\downarrow)$ to $d_i(v)$. However
the letters in $A^i(u)\!\!\uparrow$ will be compared to letters
that lie in either $C^i(u)\!\!\downarrow$, $B^i(u)\!\!\uparrow$, or later
letters in $A^i(u)\!\!\uparrow$, and hence they will contribute
0 to $d_i(v)$. Thus
\begin{eqnarray*}
d_i(v)+\Sigma(s_i(v))&=& \Sigma(B^i_1(u)) - \Sigma(D^i(u)\!\!\downarrow) +
\Sigma(B^i_2(u)) + \Sigma(C^i(u)\!\!\downarrow) + \Sigma(D^i(u)\!\!\downarrow) \\
&=& \Sigma(B^i(u)\!\!\uparrow) + \Sigma(C^i(u)\!\!\downarrow) =\Sigma(s_i(u)).
\end{eqnarray*}
\ \\
{\bf Case 3.} $v_i \in C^i(u)\!\!\downarrow$.\\
\ \\
In this case $a_i +b_i ** i\}$. Thus any letter in
$B^i(u)\!\!\uparrow C^i_1(u)$ is greater than or equal to every letter in $D^i(u)\!\!\downarrow$
so that such letters will contribute
$\Sigma(B^i(u)\!\!\uparrow) + \Sigma(C^i_1(u)) - \Sigma(D^i(u)\!\!\downarrow)$ to $d_i(v)$. However
the letters in $A^i(u)\!\!\uparrow$ will be compared to letters
that lie in either $C^i(u)\!\!\downarrow$, $B^i(u)\!\!\uparrow$, or later
letters in $A^i(u)\!\!\uparrow$, and hence they will contribute
0 to $d_i(v)$. Thus
\begin{eqnarray*}
d_i(v)+\Sigma(s_i(v))&=& \Sigma(B^i(u)\!\!\uparrow) + \Sigma(C^i_1(u)) - \Sigma(D^i(u)\!\!\downarrow) +
\Sigma(C^i_2(u)) + \Sigma(D^i(u)\!\!\downarrow) \\
&=& \Sigma(B^i(u)\!\!\uparrow) + \Sigma(C^i(u)\!\!\downarrow) =\Sigma(s_i(u)).
\end{eqnarray*}
\ \\
{\bf Case 4.} $v_i \in D^i(u)\!\!\uparrow$.\\
\ \\
In this case $a_i +b_i +c_i ** 7$.
The sequences of coefficients in the expansion
of $A(151;1,x)$ and $A(161;1,x)$ starting at $x$ are \seqnum{A145112} and \seqnum{A145113} respectively, and
count the number of length $n$ binary words with
fewer than four (resp. five) 0-digits between any pair of consecutive 1-digits.
In general, the sequence of coefficients in the expansion of $A(1i1;1,x)$ for $i\geq2$ starting at $x$ counts
the number of length $n$ binary words with fewer than $i-1$ 0-digits between any pair of consecutive 1-digits.
We then obtain a new combinatorial interpretation of these numbers as the number of words of $u$ such that $\sum (u) =n+1$
and $u$ does not embed $1i1$. There is a simple bijective proof of this fact. Given a binary word $w$ of length $n$ with
fewer than $i-1$ 0-digits between any pair
of consecutive 1-digits, we define $f(w)$ to be the word obtained by adding a 1 to the end of $w$ and then replacing
every maximal run of $k$ consecutive 0-digits followed by a 1 by the single digit $k+1$. For example,
suppose $w=01000111010000100100$ then $$f(w)=2~4~1~1~2~5~3~3.$$ Note that $\sum f(w)=n+1$ and the condition of having fewer
than $i-1$ 0-digits in $w$ guarantees that $f(w) \in \mathcal{A}(1i1)$. It is a simple verification that $f$ is indeed
a bijection between length $n$ binary words having fewer than $i-1$ 0-digits between any pair of consecutive 1-digits and
words of weight $n+1$ in $\mathcal{A}(1i1).$
Finally, we note that as in the previous subsection, we can also compute the coefficients in the expansion of $F(1~(1+s)~1;1,x)$ for various $s$. However, these do not appear in the OEIS, and therefore we omit them here.
\subsection{The word 123}
Another simple example is $S(123;t,x)$. In this case
it is easy to see that $d_1(123) = d_2(123) = d_3(123) =0$.
Thus it follows that
\begin{eqnarray*}
S(123;t,x) &=&
\frac{t^3x^6}{t^3x^6+(1-x-xt)(t^2x^5+tx^3(1-x)+(1-x)^2)} \ \mbox{and} \\
A(123;t,x) &=& \frac{1-x}{1-x-xt}(1 -S(123;t,x)).
\end{eqnarray*}
One can compute that
\begin{eqnarray*}
S(123;t,x) &=&
\frac{x^6}{(1-x)^2(x^4-x^3+2x-1)} \ \mbox{and} \\
A(123;t,x) &=& \frac{1-2x+x^2+x^3-x^4+x^5}{(1-x)^2(x^4-x^3+2x-1)}.
\end{eqnarray*}
Expanding these functions as power series about $x=0$ and letting $t=1$, we obtain
that
\begin{eqnarray*}
S(123;1,x) &=& x^6+4x^7+11x^8+25x^9+52x^{10}+103x^{11} +199x^{12}+\cdots
\mbox{and} \\
A(123;1,x) &=& 1+x+2 x^2+4 x^3+8 x^4+16 x^5+31 x^6+59 x^7+111 x^8+\\
&&208 x^9+389 x^{10}+727 x^{11}+1358 x^{12}+\cdots.
\end{eqnarray*}
In this case, neither sequence of coefficients have appeared in
the OEIS.
\section{Wilf equivalence for words of length 3} \label{sec:3}
We now turn to the classification of the Wilf equivalence classes for all words of length
3 in $\mathcal{P}_1 = (\mathbb{P}, \leq)$. Theorem~\ref{thm:S} will provide the necessary information
for words with increasing/decreasing factorizations.
The only words of length 3 without increasing/decreasing
factorizations are words of the form $bac$ or $cab$ where $a < b \leq c$.
But Kitaev, Liese, Remmel, and Sagan (Lemma 4.1 of \cite{KLRS}) show that
any word is Wilf equivalent to its reverse, so it suffices to consider $bac$. To that end, we
give an explicit formula
for $S(bac;t,x)$ where $a < b \leq c$.
\begin{theorem}\label{thm:bac} For positive integers $a < b \leq c$,
$$
S(bac;t,x) = \frac{t^3 x^{a+b+c} (1 + t x^c(1 + x + \cdots + x^{b-a-1}))}
{(1-x-tx)\psi_{a,b,c}(t,x) + t^3 x^{a+b+c} (1 + t x^c(1 + x + \cdots + x^{b-a-1})) }
$$
where
$$
\psi_{a,b,c}(t,x) = (1-x)^2 + t x^c(1-x) + t^2 x^{a+c} + t^3 x^{a+2c}(1 + x + \cdots +
x^{b-a-1}).
$$
\end{theorem}
\begin{proof} We start with the following expression, which follows from (\ref{eq:S2}) and Lemma~\ref{lem:S},
with extra terms to account for the fact that $bac$ does not have an increasing/decreasing
factorization:
\begin{eqnarray*}
S(bac;t,x) & = & A(bac;t,x) \: t^3 \frac{x^{a+b+c}}{(1-x)^3} - S(bac;t,x)\: t^2 \frac{x^{a+c}}{(1-x)^2}
-S(bac;t,x)\: t \frac{x^c}{1-x}\\
&& + A(bac;t,x) \: t^4 \frac{x^{b+2c}}{(1-x)^3} (x^a + \cdots + x^{b-1})\\
&&- S(bac;t,x) \: t^3 \frac{x^{2c}}{(1-x)^2} (x^a + \cdots + x^{b-1}).\\
\end{eqnarray*}
The first term,
$$
A(bac;t,x) \: t^3 \frac{x^{a+b+c}}{(1-x)^3},
$$
is the generating function for words consisting of an embedding of $bac$ appended to
a word that avoids $bac$. This includes the words in $\S(bac)$, but also includes
words that end in overlapping embeddings of $bac$, either in the last four or five characters
(and do not embed $bac$ prior to those embeddings). So we need to remove
the terms associated with these words.
First consider words $w$ that end in overlapping embeddings in the last five characters, as shown:
$$
\begin{array}{rlllllll}
w= & \cdots & b^+ & a^+ & c^+ & a^+ & c^+ \\
\hline
&& b & a & c & & \\
& & & & b & a & c
\end{array},
$$
where for a positive integer $m$, $m^+$ represents any integer greater than or equal
to $m$.
Since $b \leq c$, these words can be formed by appending an embedding of $ac$ to words in $\S(bac)$.
So the second term on the right hand side,
$$
S(bac;t,x)\: t^2 \frac{x^{a+c}}{(1-x)^2},
$$
accounts for these words.
Now consider words that end in overlapping embeddings in the final four characters only:
$$
\begin{array}{rlllllll}
w= & \cdots & b^+ & b^+ & c^+ & c^+ \\
\hline
&& b & a & c & & \\
& & & b & a & c
\end{array}.
$$
The third term,
$$
S(bac;t,x)\: t \frac{x^c}{1-x},
$$
removes the terms associated with these words by appending an embedding of $c$ to words
in $\S(bac)$. However, because $a < b$, this
also includes terms associated with
words of the following form:
$$
\begin{array}{rlllllll}
w= & \cdots & b^+ & [a,b) & c^+ & c^+ \\
\hline
&& b & a & c & & \\
& & & b & a & c
\end{array},
$$
that is, words that first embed $bac$ starting at the fourth character
from the end, and end in an embedding of $bacc$ but not $bbcc$.
To correct for these terms, consider the fourth term on the right hand side:
$$
A(bac;t,x) \: t^4 \frac{x^{b+2c}}{(1-x)^3} (x^a + \cdots + x^{b-1}).
$$
This is the generating function for those words that end in an embedding
of $bacc$ but not $bbcc$ (hence the $x^a + \cdots + x^{b-1}$ term),
appended to words that avoid $bac$. So this includes the
words that we want, but also may include words that first
embed $bac$ beginning at the fifth or sixth character from the end.
However, the first of these situations is impossible, as shown,
$$
\begin{array}{rlllllll}
w= & \cdots & \bullet & b^+ & [a,b) & c^+ & c^+ \\
\hline
&& b & a & c \\
&&& b & a & c & & \\
& & & & b & a & c
\end{array},
$$
since a character in $[a,b)$ cannot embed $c$. The final term,
$$
S(bac;t,x) \: t^3 \frac{x^{2c}}{(1-x)^2} (x^a + \cdots + x^{b-1}),
$$
accounts for the second possibility,
$$
\begin{array}{rlllllllll}
w= & \cdots & b^+ & a^+ & c^+ & [a,b) & c^+ & c^+ \\
\hline
&& b & a & c \\
&&&& b & a & c & & \\
& && & & b & a & c
\end{array},
$$
by appending an embedding of $acc$, whose first character is in $[a,b)$, to
words in $\S(bac)$.
We can now solve for $S(bac;t,x)$:
\begin{eqnarray*}
S(bac;t,x) & =& \frac{t^3 \frac{x^{a+b+c}}{(1-x)^3} + t^4 \frac{x^{b+2c}}{(1-x)^3} (x^a + \cdots + x^{b-1})}{1+ t \frac{x^c}{1-x} + t^2 \frac{x^{a+c}}{(1-x)^2} + t^3 \frac{x^{2c}}{(1-x)^2} (x^a + \cdots + x^{b-1})} A(bac;t,x) \\ \\
& = & \frac{t^3 x^{a+b+c} + t^4 x^{b+2c}(x^a + \cdots + x^{b-1})}
{(1-x)^3 + t x^c (1-x)^2 + t^2 x^{a+c} (1-x) + t^3 x^{2c} (1-x) (x^a + \cdots + x^{b-1})} \\\\
&& \cdot A(bac;t,x).
\end{eqnarray*}
Substituting $A(bac;t,x)=\frac{1-x}{1-x-tx} (1-S(bac;t,x))$, we obtain
\begin{eqnarray*}
S(bac;t,x) & =&\frac{t^3 x^{a+b+c} + t^4 x^{b+2c}(x^a + \cdots + x^{b-1})}
{(1-x)^2 + t x^c (1-x) + t^2 x^{a+c}+ t^3 x^{2c} (x^a + \cdots + x^{b-1})}\\
&& \cdot \frac{1}{1-x-tx} (1-S(bac;t,x)).
\end{eqnarray*}
Solving for $S(bac;t,x)$ and factoring appropriate terms gives the result.
\end{proof}
As a corollary, we can now classify Wilf equivalence for words of the form $bac$ with $a < b \leq c$.
%_______________________________________
\begin{corollary}\label{cor:bac}
For positive integers $a < b \leq c$, the only words
Wilf equivalent to $bac$ are $bac$ and
$cab$.
\end{corollary}
\begin{proof} As noted previously, $bac \backsim cab $ since they are reverses of each other. So it remains
to show that no other words of length three are Wilf equivalent to these two.
First, note that $1 + t x^c (1+x + \cdots + x^{b-a-1})$ does not
divide the denominator of the expression for $S(bac;t,x)$ in
Theorem~\ref{thm:bac}
since, for example, it does not divide it when $x=1$. So $S(bac;t,x)$ does not have
a single monomial in the numerator, and therefore is not equal to $S(u;t,x)$ for any $u$ that
has an increasing/decreasing factorization by Theorem~\ref{thm:S}. So
suppose $bac \backsim b'a'c'$ with $a' < b' \leq c'$. We will show that $a = a'$, $b=b'$
and $c = c'$.
Equating $S(bac;t,x)$ and
$S(b'a'c';t,x)$ from Theorem~\ref{thm:bac}, we have
\begin{eqnarray*}
&&\phi_{a,b,c}(t,x) \left[ (1-x-tx)\psi_{(a',b',c')}(t,x) + \phi_{a',b',c'}(t,x)\right]\\
&& = \phi_{a',b',c'}(t,x) \left[ (1-x-tx)\psi_{(a,b,c)}(t,x) + \phi_{a,b,c}(t,x)\right],
\end{eqnarray*}
where $\phi_{a,b,c}(t,x) = t^3 x^{a+b+c}(1+tx^c(1+x+ \cdots + x^{b-a-1}))$,
and similarly for $\phi_{a',b',c'}(t,x)$.
Since $bac \backsim b'a'c'$, we have $a+b+c = a' + b' + c'$, so we may simplify to
\begin{eqnarray*}
&& (1+tx^c(1+x+ \cdots + x^{b-a-1}))\psi_{(a',b',c')}(t,x) \\
&& = (1+tx^{c'}(1+x+ \cdots + x^{b'-a'-1}))\psi_{(a,b,c)}(t,x).
\end{eqnarray*}
Recalling that
$$ \psi_{(a,b,c)}(t,x) = (1-x)^2 + t x^c(1-x) + t^2 x^{a+c} + t^3 x^{2c}(x^a + \cdots +
x^{b-1}), $$
and equating powers of $t^2$ on both sides, we have
$$
x^{a'+c'}+x^{c+c'}(1-x)(1+x+ \cdots + x^{b-a-1}) =
x^{a+c}+x^{c+c'}(1-x)(1+x+ \cdots + x^{b'-a'-1})
$$
Since $a b$
\item $\{bac, cab\}$ and $\{abc, acb, cba, bca\}$ if $a < b < c$.
\end{enumerate}
\end{theorem}
\begin{proof} Theorem~\ref{thm:rearrangements} establishes the equivalence among words in the sets $\{aab,aba,baa\}$ in case 2, the sets $\{aab, baa\}$ in
case 3, and $\{abc, acb, cba, bca\}$ in case 4, as well as the fact that these sets are all in distinct Wilf equivalences classes.
The remaining cases, $\{bac, cab\}$ in case 4 and $\{aba\}$ in case 3, follow from Corollary~\ref{cor:bac}.
\end{proof}
Note that when we consider the permutations of $S_3$, there are only
two Wilf equivalence classes, namely, $\{123,132,321,231\}$ and
$\{213,312\}$. We computed $S(123;t,x)$ and $A(123;t,x)$
in the previous section. Thus to complete the possibilities
for $S(\sg;t,x)$ for $\sg \in S_3$, we need only compute
$S(213;t,x)$ and $A(213;t,x)$. In this case, we must use
Theorem \ref{thm:bac} from which we obtain that
\begin{eqnarray*}
S(213;t,x)&=& \frac{t^3x^6(1+tx^3)}{(1-x-xt)((1-x)^2 +tx^3(1-x)+t^2x^4+t^3x^7)+t^3x^6(1+tx^3)} \ \mbox{and} \\
A(213;t,x)&=& \frac{1-x}{1-x-xt}(1-S(213;t,x)).
\end{eqnarray*}
One can compute that
\begin{eqnarray*}
S(213;1,x)&=& \frac{x^6+x^8}{1-4x+5x^2-x^3-2x^4+2x^6+x^7-2x^8} \ \mbox{and}\\
A(213;1,x)&=& \frac{(1-x)(1-4x+5x^2-x^3-2x^4+x^6+x^7-3x^8)}
{(1-2x)(1-4x+5x^2-x^3-2x^4+2x^6+x^7-2x^8)}.
\end{eqnarray*}
In this case, if one expands these functions as power series
about $x=0$, one obtains
\begin{eqnarray*}
S(213;1,x)&=& x^6+4 x^7+11 x^8+26 x^9+55 x^{10}+109 x^{11}+207 x^{12}+381 x^{13}+\\
&&684 x^{14}+1201 x^{15}+O[x]^{16} \ \mbox{and}\\
A(213;1,x)&=& 1+x+2 x^2+4 x^3+8 x^4+16 x^5+31 x^6+59 x^7+111 x^8+207 x^9+\\
&&385 x^{10}+716 x^{11}+1334 x^{12}+2494 x^{13}+4685 x^{14}+8853 x^{15}
+O[x]^{16}.
\end{eqnarray*}
However, neither of the two sequence of coefficients have appeared
in the OEIS.
\section{The strong rearrangement conjecture} \label{sec:rearrangement}
In this section we discuss the strong rearrangement
conjecture and its connection to the family
of finite posets
$\mathcal{P}_{[m]} = ([m]^*, \leq)$. We also give an analogue of Theorem
\ref{thm:S} for $S(u;x_1, \ldots ,x_n)$.
Our first result relates Wilf equivalence in $[m]^*$ to Wilf equivalences in $\mathbb{P}^*$ that are witnessed by rearrangement maps.
\begin{theorem}\label{thm:lift} Suppose $u,v \in [m]^*$ for some positive integer
$m$. Then $u \backsim_{[m]} v$ if and only if there exists a rearrangement map $f: \mathbb{P}^* \rightarrow \mathbb{P}^*$ that witnesses the Wilf equivalence $u \backsim v$.
\end{theorem}
\begin{proof} First note that if there is a rearrangement map $f: \mathbb{P}^* \rightarrow \mathbb{P}^*$ that witnesses the Wilf equivalence $u \backsim v$, then the restriction of $f$ to $[m]^*$ is a $W_{[m]}$-preserving bijection that shows $u \backsim_{[m]} v$.
For the converse, suppose $u, v \in [m]^*$ and $u \backsim_{[m]} v$, so that
$F(u; x_1, \ldots , x_m) = F(v; x_1, \ldots , x_m)$. Then
there is a $W_{[m]}$-preserving
bijection $g:\F(u) \cap [m]^* \rightarrow \F(v) \cap [m]^*$.
So $g(w)$ is a rearrangement of $w$ for all $w$.
This bijection
can then be lifted to the desired rearrangement $f$, as follows. Suppose
$w = w_1 \cdots w_n \in \P^*$ and $1 \leq i_1 < \cdots < i_l \leq n$ is
the sequences of indices $i$ such that $w_i \geq m$. Then let
$\overline{w}$ be the word in $[m]^*$ that results by replacing each
$w_{i_k}$ with $m$. Then $u \leq w$ if, and only if, $u \leq \overline{w}$.
Now apply $g$ to $\overline{w}$. Then since $z = g(\overline{w})$ is a rearrangement
of $\overline{w}$, there is a sequence $1 \leq j_1 < \cdots < j_l \leq n$
consisting of all the indices $j$ such that $z_j = m$. Then let $f(w)$ be
the result of replacing $z_{j_k}$ by $w_{i_k}$ for $k = 1, \ldots, l$.
\end{proof}
Theorem~\ref{thm:lift} shows that the question of whether
$u \backsim v$ implies a rearrangement witnessing the equivalence
can be answered by restricting to a finite alphabet. We have computed $S(u;x_1,\ldots ,x_5)$ for all permutations in $S_n$ for $n \leq 5$ and indeed, if $u \backsim v$ in this case, then
$S(u;x_1,\ldots ,x_5) = S(v;x_1,\ldots ,x_5)$. Thus the
strong rearrangement conjecture holds for these words.
Next we consider an analogue of Theorem~\ref{thm:S} for the more refined
generating functions $S(u;x_1, \ldots ,x_m)$. It is still the case
that
\begin{equation}\label{eq:Sx2}
\S(u) \cap [m]^* = (\A(u) \cap [m]^*)(\mathcal{W}(u)\cap [m]^*)-
\left(\bigcup_{i=1}^{n-1} (\S^{(i)}(u) \cap [m]^*)\right).
\end{equation}
It is easy to see that
\begin{eqnarray}\label{eq:Sx3}
\sum_{w \in \A(u)\W(u) \cap [m]^*}W_{[m]}(w) &=&
A(u;x_1,\ldots,x_m)\prod_{r=1}^n \sum_{s=u_i}^m x_j \nonumber \\
&=& \frac{1}{1-\sum_{i=1}^m x_i}(1-S(u;x_1, \ldots, x_m))\prod_{r=1}^n
\sum_{s=u_r}^m x_s.
\end{eqnarray}
We also have that
\begin{equation*}\label{eq:Sx4}
\S^{(i)}(u)\cap [m]^* = \bar{\S}^{(i)}(u)\W(s_i(u)) \cap [m]^*.
\end{equation*}
Thus if
\begin{eqnarray*}
S^{(i)}(u,x_1, \ldots, x_m) &=& \sum_{w \in \S^{(i)}(u) \cap [m]^*} W_{[m]}(w) \ \mbox{and} \\
\bar{S}^{(i)}(u,x_1, \ldots, x_m) &=& \sum_{w \in \bar{\S}^{(i)}(u) \cap [m]^*} W_{[m]}(w),
\end{eqnarray*}
then we will have
\begin{equation*}\label{eq:Sx5}
S^{(i)}(u,x_1, \ldots, x_m) = \bar{S}^{(i)}(u,x_1, \ldots, x_m)\prod_{r=i+1}^n
\sum_{s=u_r}^m x_s.
\end{equation*}
The only step in our proof of Theorem \ref{thm:S} which does not
have an analogue in this case is the fact that
$$\bar{S}^{(i)}(u;t,x) = x^{d_i(u)} S(u;t,x).$$
It will no longer be the case that
$\bar{S}^{(i)}(u;x_1, \ldots,x_m)$ is a multiple of $S(u;x_1, \ldots, x_m)$ if
$d_i(u) > 0$. However, if $d_i(u) =0$, then it will be the case
that $\bar{S}^{(i)}(u) \cap [m]^* = \S^{(i)}(u) \cap [m]^*$ so that
\begin{equation*}\label{eq:Sx6}
\bar{S}^{(i)}(u,x_1, \ldots, x_m) =S(u,x_1, \ldots, x_m).
\end{equation*}
Thus if $d_i(u) =0$ for all $i =1, \ldots, n-1$, then
we will have
\begin{equation}\label{eq:Sx7}
S^{(i)}(u,x_1, \ldots, x_m) = S(u,x_1, \ldots, x_m)\sum_{r=i+1}^n
\sum_{s=u_r}^m x_s
\end{equation}
for all $i$. However, it is easy to see that $d_i(u) =0$ for all $i =1, \ldots, n-1$ if and only if $u_1 \leq \cdots \leq u_n$. In that case,
we can see from (\ref{eq:Sx2}), (\ref{eq:Sx3}), and
(\ref{eq:Sx7}) that
\begin{eqnarray*}\label{eq:Sx8}
S(u,x_1, \ldots, x_m) &=& \frac{1}{1-\sum_{i=1}^m x_i}(1-S(u;x_1, \ldots, x_m))\sum_{r=1}^n \sum_{s=u_r}^m x_s \\
&&- \sum_{i=1}^{n-1} S(u,x_1, \ldots, x_m)\sum_{r=i+1}^n
\sum_{s=u_r}^m x_s.
\end{eqnarray*}
Solving for $S(u,x_1, \ldots, x_m)$ will then result in the following
theorem.
\begin{theorem} \label{thm:Sfinite}
Suppose $u = u_1 \ldots u_n \in [m]^*$ is weakly increasing. Then
\begin{eqnarray*}
S(u;x_1, \ldots, x_m) & = & \frac{\prod_{i=1}^n\sum_{j=u_i}^m x_j}
{\left(1 + \sum_{i=1}^{n-1} \prod_{j=i+1}^n\sum_{l=u_j}^m x_l\right)
( 1 -\sum_{i=1}^m x_i ) + \prod_{i=1}^n\sum_{j=u_i}^m x_j}.
\end{eqnarray*}
\end{theorem}
\section{Further work} \label{sec:conclusion}
Many of the ideas in this paper can be extended to generalized factor order on $\mathbb{P}^*$ with
other partial orders. In particular, in \cite{LLR} we consider the mod $k$ partial order on $\mathbb{P}^*$ defined by setting $m \leq_k n$ if $m \leq n$ and $m = n \text{ mod } k$. For example, the Hasse diagram for the mod 3 partial order consists of the three chains
\begin{eqnarray*}
&&1 \leq_3 4 \leq _3 7 \leq_3 \cdots, \\
&&2 \leq_3 5 \leq _3 8 \leq_3 \cdots, \text{ and}\\
&&3 \leq_3 6 \leq _3 9 \leq_3 \cdots.\
\end{eqnarray*}
An generalization of Theorem~\ref{thm:S} in this context applies to a rich class of words that generalizes the set of words in $\mathbb{P}^*$ with increasing/decreasing factorizations. One interesting result of this theorem is that the rearrangement conjectures do not hold in general for the mod $k$ partial order with $k \geq 2$, however we can identify those words for which we believe the rearrangement conjectures do hold. We refer the reader to \cite{LLR} for details.
\begin{thebibliography}{25}
\bibitem{EM} E. Egge and T. Mansour, 132-avoiding two-stack sortable
permutations, Fibonacci numbers, and Pell numbers, \textit{Discrete Appl. Math.}
{\bf 143} (2004), 72--83.
\bibitem{KLRS} Sergey Kitaev, Jeffrey Liese, Jeffrey Remmel, Bruce E.
Sagan, Rationality, irrationality, and Wilf equivalence in generalized
factor order. \textit{Electron. J. Combin.} \textbf{16} (2) (2009),
\href{http://www.combinatorics.org/Volume_16/Abstracts/v16i2r22.html}{Paper R22}.
\bibitem{SV} Bruce E. Sagan and Vincent Vatter, The M\"{o}bius function of a composition poset. \textit{J. Algebraic Combin.} \textbf{24} (2006), 111--136.
\bibitem{Wilf} H. S. Wilf, The patterns of permutations.
\textit{Discrete Math.} \textbf{257} (2002), 575--583.
\bibitem{LLR} Thomas Langley, Jeffrey Liese, and Jeffrey Remmel, Wilf
equivalence for generalized factor orders modulo $k$, preprint.
\end{thebibliography}
\bigskip
\hrule
\bigskip
\noindent 2000 {\it Mathematics Subject Classifications}:
05A15, 68R15, 06A07
\noindent \emph{Keywords: }
composition, factor orders,
generating function, partially ordered set, rationality, Wilf equivalence
\bigskip
\hrule
\bigskip
\noindent (Concerned with sequences
\seqnum{A000045},
\seqnum{A000071},
\seqnum{A000073},
\seqnum{A000078},
\seqnum{A000124},
\seqnum{A000126},
\seqnum{A000292},
\seqnum{A001591},
\seqnum{A001949},
\seqnum{A007800},
\seqnum{A008466},
\seqnum{A008937},
\seqnum{A014162},
\seqnum{A050231},
\seqnum{A050232},
\seqnum{A050233},
\seqnum{A107066},
\seqnum{A145112},
\seqnum{A145113}, and
\seqnum{A172119}.)
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received May 24 2010;
revised version received March 16 2011.
Published in {\it Journal of Integer Sequences}, March 26 2011.
\bigskip
\hrule
\bigskip
\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in
\end{document}
*