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%%%%% Auteur
\author{\firstname{Sheila} \lastname{Sundaram}}
\address{Pierrepont School\\
One Sylvan Road North\\
Westport\\
CT 06880, USA}
\email{shsund@comcast.net}
%%%%% Sujet
\keywords{Subword order, reflection representation, $h$-positivity, Whitney homology, Kronecker product, internal product, Stirling numbers.}
\subjclass{05E10, 20C30}
%%%%% Gestion
\DOI{10.5802/alco.184}
\datereceived{2020-07-15}
\daterevised{2021-05-09}
\dateaccepted{2021-05-12}
%%%%% Titre et résumé
\title[Homology of subword order]{The reflection representation in the homology of subword order}
\begin{abstract}
We investigate the homology representation of the symmetric group on rank-selected subposets of subword order.
We show that the homology module for words of bounded length, over an alphabet of size $n,$
decomposes into a sum of tensor powers of the $S_n$-irreducible $S_{(n-1,1)}$ indexed by the partition $(n-1,1),$ recovering, as a special case, a theorem of Bj{\"o}rner and Stanley for words of length at most $k.$ For arbitrary ranks we show that the homology is an integer combination of positive tensor powers of the reflection representation $S_{(n-1,1)}$, and conjecture that this combination is nonnegative. We uncover a curious duality in homology in the case when one rank is deleted.
We prove that the action on the rank-selected chains of subword order is a nonnegative integer combination of tensor powers of $S_{(n-1,1)}$, and show that its Frobenius characteristic is $h$-positive and supported on the set
$T_{1}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 1\}.$
Our most definitive result describes the Frobenius characteristic of the homology for an arbitrary set of ranks, plus or minus one copy of the Schur function $s_{(n-1,1)},$ as an integer combination of the set
$T_{2}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$ We conjecture that
this combination is nonnegative, establishing this fact for particular cases.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\maketitle
%
\section{Introduction}
Let $A^*$ denote the free monoid of words of finite length in an alphabet $A.$
Subword order is defined on $A^*$ by setting $ u\le v$ if $u$ is a subword of $v,$ that is,
the word $u$ is obtained by deleting letters of the word $v.$ This makes $(A^*, \leq)$ into a graded poset
with rank function given by the length $|w|$ of a word $w,$ the number of letters in $w$. The topology of this poset was first studied by Farmer (1979)
and then by Bj{\"o}rner, who showed in~\cite[Theorem~3]{Bj1} that any interval of this poset admits a dual CL-shelling. The intervals are thus homotopy Cohen--Macaulay, as well as all rank-selected subposets
obtained by considering only words whose rank belongs to
a finite set $S$~\cite[Theorem~4.1]{BjShellableTAMS}, \cite[Theorem~8.1]{BjWachs}.
Suppose now that the alphabet $A$ is finite, of cardinality $n$. The symmetric group
$S_n $ acts on $A$, and thus on $A^*.$ To avoid trivialities we will assume $n\ge 2.$
In this paper we describe the homology representation
of intervals $[r, k]$ of consecutive ranks in $A^*$, as well as some other rank-selected subposets, using the Whitney homology technique and other methods developed in~\cite{Su0}. All homology in this paper is taken over the field of complex numbers.
We refer the reader to~\cite{St1} for general facts about rank-selection.
We show that the unique nonvanishing
homology of the rank-selected subposet $A^*_{[r,k]}$
decomposes as a direct sum of copies of $r$ consecutive tensor powers of the reflection
representation of $S_n$, that is, the irreducible representation $S_{(n-1,1)}$ indexed by the partition $(n-1,1).$ Theorem~\ref{Bordeaux1991mai} on consecutive ranks generalises a theorem in~\cite{Bj1} (conjectured by Bj{\"o}rner and proved by Stanley) on the homology representation of the poset of all words of length at most $k.$ We establish similar results for the Whitney and dual Whitney homology modules. Both turn out to be permutation modules in each degree, with pleasing orbit stabilisers. Theorem~\ref{rank-deletion} establishes the nonnegativity property with respect to tensor powers of $S_{(n-1,1)}$ for the case when one rank is deleted from the interval $[1,k],$ and leads to a curious homology isomorphism (Proposition~\ref{rank-deletion-duality}), suggesting a homotopy equivalence between the simplicial complexes associated to the rank sets $[1,k]\backslash {\{r\}}$ and $[1,k]\backslash {\{k-r\}},$ for fixed $r, 1\le r\le k-1.$ Finally Theorem~\ref{TwoRanks} establishes that the homology is a nonnegative sum of tensor powers of $S_{(n-1,1)}$ for rank-sets of size 2.
More generally, we show in Theorem~\ref{RankSelectedChains} that for any nonempty subset $S$ of ranks $[1,k],$ the homology representation of $S_n$ may be written as an integer combination of positive tensor powers of the reflection representation. We propose the following conjecture, which is supported by Theorems~\ref{Bordeaux1991mai}, \ref{rank-deletion} and~\ref{TwoRanks}:
\begin{conj}\label{Conj:tensorpowers} Let $A$ be an alphabet of size $n\ge 2.$ Then the $S_n$-homology module of any finite nonempty rank-selected subposet of subword order on $A^*$ is a \emph{nonnegative} integer combination of positive tensor powers of the irreducible indexed by the partition $(n-1,1).$
\end{conj}
These considerations lead us to examine the tensor powers of the reflection representation (see Section~\ref{PowersOfRefl}), and the question of how many tensor powers are linearly independent characters. In answering these questions, we are led to a decomposition (Theorem~\ref{ReflRepMain}) showing that the $k$th tensor power of $S_{(n-1,1)}$ plus or minus one copy of $S_{(n-1,1)},$ has Frobenius characteristic equal to a nonnegative integer combination of the homogeneous symmetric functions $\{h_{(n-r, 1^r)}: r\ge 2\}.$ It is ``almost'' an $h$-positive permutation module. (In general the homology itself is \emph{not} a permutation module.) Inspired by this phenomenon, we prove, in Theorem~\ref{Homologyhbasis}, that in fact for all rank subsets $T,$
the homology module $\tilde{H}(T)$
has the property that $\tilde{H}(T) + (-1)^{|T|} S_{(n-1,1)} $ has Frobenius characteristic equal to an \emph{integer} combination of the homogeneous symmetric functions $\{h_{(n-r, 1^r)}: r\ge 2\}.$ Theorem~\ref{AlmosthpositiveInstances} establishes the truth of the following conjecture for
the homology of several subsets of ranks.
%
\begin{conj}\label{Conj:almost-h-pos}
Let $A$ be an alphabet of size $n\ge 2.$ Then the $S_n$-homology module of any finite nonempty rank-selected subposet of subword order on $A^*$, plus or minus one copy of the reflection representation of $S_n,$ is a permutation module. In fact its Frobenius characteristic is $h$-positive and supported on the set $T_{2}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$
\end{conj}
We give a simple criterion for when Conjecture~\ref{Conj:tensorpowers} will imply Conjecture~\ref{Conj:almost-h-pos} in Lemma~\ref{RefRephPos}.
The main results of this paper are summarised below. Let $A$ be an alphabet of size $n\ge 2,$ and $T\subseteq [1,k]$ a subset of ranks.
\begin{theo}\label{SummaryChains} The $S_n$-module induced by the action of $S_n$ on the maximal chains of the rank-selected subposet of $A^*$ of words with lengths in $T,$ is a \textegras{nonnegative} integer combination of tensor powers of the reflection representation $S_{(n-1,1)}.$ If $|T|\ge 1,$ this module has $h$-positive Frobenius characteristic supported on the set $T_{1}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 1\}.$
\end{theo}
\begin{theo}\label{Summary} The homology module $\tilde{H}(T)$ of words with lengths in $T$ is an integer combination of positive tensor powers of the reflection representation $S_{(n-1,1)},$
with the property that $\tilde{H}(T) + (-1)^{|T|} S_{(n-1,1)} $ has Frobenius characteristic equal to an integer combination of the homogeneous symmetric functions $\{h_{(n-r, 1^r)}: r\ge 2\}.$
Both integer combinations are \textegras{nonnegative}
when $T$ is one of the following rank sets:
%
$ (1)\ [r,k], k\ge r\ge 1; \quad (2)\ [1,k]\backslash \{r\}, k\ge r\ge 1;\quad
(3)\ \{1\le s_10.$ It is easy to verify similarly that the reduced Stirling numbers $S^*(j,d)$ satisfy the recurrence
$ S^*(n+1,d)=S^*(n,d-1)+(d-1)S^*(n,d),$ by examining the possibilities for inserting $(n+1)$ into a partition of $[n]$ into $d$ blocks. A comparison of the recurrences immediately shows that in fact
\[
S^*(n+1,d)=S(n,d-1)\quad \text{for all } n\ge 0,d\ge 1.
\]
See~\cite{Munagi} for generalisations of these numbers.
%
Recall that in Theorem~\ref{WHsubword}, the $j$th Whitney homology of $A^*_{n,k}$ $j\ge 2,$ was determined as a sum of two consecutive tensor powers of $S_{(n-1,1)}.$ From Lemma~\ref{surj-normal-to-ptns} and Proposition~\ref{LowerIntTopo} we now have the following surprising result.
\begin{prop}\label{WhPermModule} Each Whitney homology module of subword order, and hence the sum of two consecutive tensor powers of the reflection representation, has $h$-positive Frobenius characteristic, and in particular it is a permutation module. We have $\ch W\!H_0=h_n, \ch W\!H_1=h_1 h_{n-1},$ and for $k\ge j\ge 2,$ the $j$th Whitney homology of $A^*_{n,k}$ has Frobenius characteristic
\begin{equation}\label{StirlingWh}
\begin{aligned}
(h_1h_{n-1})*s_{(n-1,1)}^{*(j-1)}&=\sum_{d= 2}^j S(j-1,d-1)\, h_1^d h_{n-d}\\
&=\sum_{d=2}^j S^*(j,d)\,h_1^d h_{n-d}= h_1(h_1h_{n-2})^{*j-1},
\end{aligned}
\end{equation}
a permutation module with orbits whose stabilisers are Young subgroups indexed by partitions of the form $(n-d,1^d), d\ge 0.$
\end{prop}
\begin{proof} From Equation~\eqref{Whitney} and Theorem~\ref{WHsubword}, for $k\ge j\ge 2,$ we have
\[
W\!H_j(A^*_{n,k})= S_{(n-1,1)}^{\otimes j} \oplus S_{(n-1,1)}^{\otimes j-1}= (S_{(n)}\oplus S_{(n-1,1)})\otimes S_{(n-1,1)}^{\otimes j-1}.
\]
Now by definition we also have
\[
WH_j(A^*_{n,k})=\sum_{x\in A^*_{n,k}, |x|=j} \tilde{H}(\hat 0,x).
\]
Proposition~\ref{LowerIntTopo} says that the sum runs over only normal words $x,$ and each homology module is trivial for the stabiliser of $x.$
Collecting the summands into orbits and using the surjection of Lemma~\ref{surj-normal-to-ptns} gives Equation~\eqref{StirlingWh}. The last expression is obtained by shifting the index in the sum:
\[
\sum_{d= 2}^j S(j-1,d-1)\, h_1^d h_{n-d}=h_1\sum_{d'=1}^{j-1} S(j-1,d')
h_1^{d'} h_{(n-1)-d'},
\]
and this equals $h_1(h_1h_{n-2})^{*j-1}$ by Lemma~\ref{Stirling}.
\end{proof}
%
Recall~\cite{M} that the homogeneous symmetric functions $h_\lambda$ form a basis for the ring of symmetric functions.
\begin{theo}\label{ReflRepMain} Fix $k\ge 1.$
The $k$th tensor power of the reflection representation
$S_{(n-1,1)}^{\otimes k},$ \ie the homology module $\tilde{H}_{k-1}(A^*_{n,k}),$ has the following property:
$S_{(n-1,1)}^{\otimes k}\oplus (-1)^{k} S_{(n-1,1)}$
is a permutation module $U_{n,k}$ whose Frobenius characteristic is $h$-positive, and is supported on the set $\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$
If $k=1,$ then $U_{n,1}=0.$
More precisely, the $k$-fold internal product $s_{(n-1,1)}^{ * k}$ has the following expansion in the basis of homogeneous symmetric functions $h_\lambda:$
\begin{equation} \label{RefReptoh}
\sum_{d=0}^n g_n(k,d) h_1^d h_{n-d},
\end{equation}
where $g_n(k,0)=(-1)^k, g_n(k,1)=(-1)^{k-1},$ and
%
\[
g_n(k,d)=\sum_{i=d}^k (-1)^{k-i} S(i-1, d-1), \text{ for }2\le d\le n.
\]
Hence
\[
s_{(n-1,1)}^{ * k}= (-1)^{k-1}s_{(n-1,1)}+\ch (U_{n,k}),
\]
where $\ch (U_{n,k})=\sum_{d=2}^n g_n(k,d) h_1^d h_{n-d}.$
The integers $g_n(k,d)$ are independent of $n$ for $k\le n,$ nonnegative for $2\le d\le k,$ and $g_n(k,d)=0 $ if $d>k.$
Also:
\begin{enumerate}
\item\label{theo7.7_1} $g_n(k,2)=\frac{1+(-1)^k}{2}.$
\item\label{theo7.7_2} $g_n(k,k-1)=\binom{k-1}{2}-1, k\le n.$
\item\label{theo7.7_3} $g_n(k,k)=1, k< n.$
\end{enumerate}
In particular the coefficient of $h_1^n$ in the expansion~\eqref{RefReptoh} of
$\ch S_{(n-1,1)}^{\otimes k}$ is
\[
\begin{cases} g_n(k,n)+g_n(k,n-1) &\textup{if } k> n,\\
\binom{n-1}{2} &\textup{if } k= n,\\
1 &\textup{if } k=n-1,\\
0 &\textup{otherwise}.
\end{cases}
\]
\end{theo}
\begin{proof} If $k=1,$ the terms for $d\ge 2$ in the summation in~\eqref{RefReptoh} vanish and thus the right-hand side equals the characteristic of the top homology.
The first statement, about the homology module $\tilde{H}_{k-1}(A^*_{n,k}),$ follows from Proposition~\ref{WhPermModule} and Theorem~\ref{WHmainSS}.
Fix $m$ and $d$ such that $m\ge d\ge 2.$ Let $\bar g(m,d)$ be the alternating sum of Stirling numbers
$\bar g(m,d)=\sum_{i=d}^m (-1)^{m-i} S(i-1, d-1).$ Note that $\bar g(m,d)$ equals
\begin{multline*}
[S(m-1,d-1)-S(m-2,d-1)]+[S(m-3,d-1)-S(m-4,d-1)] +\cdots \\
\cdots +
\begin{cases}
[S(d+1,d-1)-S(d,d-1)] +S(d-1,d-1), & m-d\text{ even},\\
[S(d,d-1)) -S(d-1,d-1)], & m-d\text{ odd}.
\end{cases}
\end{multline*}
Since $S(n,d)$ is an increasing function of $n\ge d$ for fixed $d,$ the coefficient $\bar g(m,d)$ is always nonnegative. It is also clear that $\bar g(m,d)= S(m-1,d-1) -\bar g(m-1,d)$ for all $m\ge d\ge 2.$
The remaining parts follow from the facts that $S(k,k-1)=\binom{k}{2}, S(k,k)=1,$ and the observation that for $k\ge n,$ the coefficient of $h_1^n$ is $g_k(k,n)+g_k(k, n-1).$ This equals $S(n-1,n-2)-1$ when $k=n.$ \end{proof}
\begin{coro}\label{TrivRepStableCase} Let $k\ge 2.$
\begin{enumerate}
\item\label{coro7.8_1} For $\min(n,k)\ge d\ge 2,$ the coefficient of $h_1^d h_{n-d}$ in $s_{(n-1,1)}^{* k}=\ch S_{(n-1,1)}^{\otimes k}$ is the nonnegative integer $g_n(k,d)$ given by the two equal expressions:
\begin{equation}\label{NewStirlingIdentity?}
\sum_{j=d}^k (-1)^{k-j} S(j-1,d-1)=\sum_{r=0}^{k-d} (-1)^r \binom{k}{k-r} S(k-r,d).
\end{equation}
In particular, when $n\ge k,$ this multiplicity is independent of $n.$
\item\label{coro7.8_2} The positive integer $\beta_n(k)=\sum_{d=2}^{\min(n,k)} g_n(k,d)$ is the multiplicity of the trivial representation in $S_{(n-1,1)}^{\otimes k}.$ When $n\ge k,$ it equals the number of set partitions $B_k^{\ge 2}$ of the set $\{1,\ldots,k\}$ with no singleton blocks. We have $\beta_n(n+1)=B_{n+1}^{\ge 2}-1$ and $\beta_n(n+2)= B_{n+2}^{\ge 2}- \binom{n+1}{2}.$
\end{enumerate}
\end{coro}
\begin{proof}
This follows from Theorem~\ref{StirlingHom} and Corollary~\ref{TrivRep}.
We have $\beta_n(n)=\sum_{d=2}^n g(n,d)=B_n^{\ge 2}= \beta_n(k)$ for $n\ge k,$ and from~\eqref{RefReptoh},
\begin{align*}
\beta_n(n+1)&=\sum_{d=2}^{n}g_{n}(n+1,d)= \sum_{d=2}^{n+1} g_{n+1}(n+1,d) - g_{n+1}(n+1,n+1)\\
&= B_{n+1}^{\ge 2}-1,\\
\beta_n(n+2) &=\sum_{d=2}^{n+2} g_{n+2}(n+2,d) - g_{n+2}(n+2,n+2)
-g_{n+2}(n+2,n+1)\\
&=B_{n+2}^{\ge 2} -1 - [\binom{n+1}{2}-1]=B_{n+2}^{\ge 2}- \binom{n+1}{2}.\qedhere
\end{align*}
%
\end{proof}
We need one final observation in order to prove Theorem~\ref{AlmosthpositiveInstances}.
\begin{lemma}\label{RefRephPos} Suppose $V$ is an $S_n$-module which can be written as an integer combination $V=\oplus_{k=1}^m c_k S_{(n-1,1)}^{\otimes k}$ of positive tensor powers of $S_{(n-1,1)}.$
Then \begin{enumerate}
\item\label{lemma7.9_1} The character value of $V$ on fixed-point-free permutations is $\sum_{k=1}^m (-1)^{k}c_k.$
\item\label{lemma7.9_2} If
$\sum_{k=1}^m (-1)^{k-1}c_k=0,$ then the Frobenius characteristic of $V$ is
supported on the set $\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$
\item\label{lemma7.9_3} If $\sum_{k=1}^m (-1)^{k-1}c_k=0$ and $c_k\ge 0$ for all $k\ge 2,$ it is $h$-positive and hence $V$ is a permutation module.
\end{enumerate}
\end{lemma}
\begin{proof} The first part follows because the value of the character of the reflection representation $S_{(n-1,1)}$ on permutations without fixed points is $(-1).$
The second and third parts are immediate from Theorem~\ref{ReflRepMain}, since we have
\[
V=\left(\sum_{k=1}^m (-1)^{k-1}c_k \right) S_{(n-1,1)} \oplus \sum_{k=2}^m c_k U_{n,k}=\sum_{k=2}^m c_k U_{n,k} ,
\]
\looseness-1
and $U_{n,k}$ is $h$-positive with support $\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$ Note that $U_{n,1}=0.$
\end{proof}
In particular from Theorem~\ref{Chains}, this gives a direct proof that the action of $S_n$ on the chains in $A_{n,k}^*$ is also a nonnegative linear combination of
$\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$
%\vskip.1in
%\noindent
%\textegras{Proof of Theorem~\ref{AlmosthpositiveInstances}:}
\begin{proof}[Proof of Theorem~\ref{AlmosthpositiveInstances}] Note that in all three cases, the module $V_T=\tilde{H}_{k-2}(A^*_{n,k}(T))+ (-1)^{|T|} S_{(n-1,1)}$ has already been shown to
be an integer combination of $k$th tensor powers of $S_{(n-1,1)},$ with nonnegative coefficients when $k\ge 2$, in Theorems~\ref{Bordeaux1991mai}, \ref{rank-deletion} and~\ref{TwoRanks}. Hence, by Lemma~\ref{RefRephPos}, it remains only to verify that the alternating sum of coefficients of the tensor powers vanishes for $V_T$ in each case.
Consider the case $T=[r,k]$. From Theorem~\ref{Bordeaux1991mai}, we must show that $(-1)^{k-r+1}$ added to the signed sum of the $(-1)^{i-1}b_i$, for the coefficients $b_i=\binom{k}{i}\binom{i-1}{ k-r},$ is zero, \ie
\begin{equation}\label{AnotherBinomialIdentity!} \sum_{i=1+k-r}^k b_i (-1)^{i-1} =(-1)^{k-r} .\end{equation}
It is easiest to use the combinatorial identity of Corollary~\ref{NewBinomialIdentity?}. Consider the two polynomials of degree $k\ge 2$ in $x$ defined by
\begin{align*}
F(x)&= \sum_{i=0}^{k-r} (-1)^i \binom{k}{r+i}(x+1)^{r+i}x^{k-(r+i)}+(-1)^{k+1-r},
\\
G(x)&= \sum_{i=1+k-r}^k \binom{k}{i}\binom{i-1}{k-r} x^i .
\end{align*}
Corollary~\ref{NewBinomialIdentity?} says $F(x)$ and $G(x)$ agree for all $x=n-1\ge 1,$ and hence $F(x)=G(x)$ identically.
In particular $F(-1)=G(-1).$ But $F(-1)= (-1)^{k+r-1}$ and
clearly $(-1) G(-1)$ is precisely the sum in the left-hand side of~\eqref{AnotherBinomialIdentity!}. The claim follows.
For the second case, $T$ is the rank-set $[1,k]\backslash \{r\},$ and from Theorem~\ref{rank-deletion} the alternating sum of coefficients in $V_T$ is clearly
\begin{equation}\label{rank-deletionIdentity}
(-1)^{k-1}+ \left[\binom{k}{r}-1\right](-1)^{k-1}+ \binom{k}{r} (-1)^{k-2}=0.
\end{equation}
For the final case, the rank set is $T=\{1\le s_1s_2-s_1}^{s_2}(-1)^{v}\sum_{j=1}^{ s_2-s_1}\binom{s_2-j}{v-j}\binom{s_1+j-1}{j}}_{\hypertarget{nameB}{(B)}}.
\]
%
Switching the order of summation, \hyperlink{nameA}{$(A)$} is equal to
\[
\sum_{j=1}^{s_2-s_1} \binom{s_1+j-1}{j}\sum_{v=j}^{s_2-s_1} \binom{s_2-j}{v-j}(-1)^{v},
\]
while \hyperlink{nameB}{$(B)$} is
\[
\sum_{j=1}^{s_2-s_1} \binom{s_1+j-1}{j}\sum_{v>s_2-s_1}^{s_2}(-1)^{v} \binom{s_2-j}{v-j}(-1)^{v}.
\]
Hence $\sum_{v=1}^{s_2} (-1)^{v} c_v$ equals
\[
\sum_{j=1}^{s_2-s_1} \binom{s_1+j-1}{j}\sum_{v=j}^{s_2} \binom{s_2-j}{v-j}(-1)^{v}=\sum_{j=1}^{s_2-s_1} \binom{s_1+j-1}{j}\sum_{w=0}^{s_2-j} \binom{s_2-j}{w}(-1)^{w-s_2},
\]
where we have put $w=s_2-v.$ But $1\le j\le s_2-s_1< s_2,$ so
the inner sum vanishes. \end{proof}
%
Note that the left-hand side of Equation~\eqref{AnotherBinomialIdentity!} is the character value on fixed-point-free permutations for the homology module $\tilde{H}(T),$ $T=[r,k]$. Hence this shows that the homology module itself cannot be a permutation module when
$k-r$ is odd, since the right-hand side then gives a value of $(-1)$ for the character.
Similarly, from Theorem~\ref{rank-deletion}, the character value on fixed-point-free permutations for $\tilde{H}(T),$ for $T=[1,k]\backslash \{r\}$,
is $\left[\binom{k}{r}-1\right](-1)^{k}+ \binom{k}{r} (-1)^{k-1},$ and this equals $(-1)^{k-1}.$ Once again we can conclude that this homology module is also not a permutation module when $k$ is even.
\begin{coro} The dual Whitney homology modules $W\!H^*_{k+1-i}(A^*_{n,k}), 1*n?$ Recall that for $k\le n$ this is the number $B_k^{\ge 2}$ of set partitions of $[k]$ with no singleton blocks, and is sequence OEIS A000296.
\end{ques}
\begin{ques} Recall that $a_{n-1}(n)=\binom{n-1}{2}.$ Is there a combinatorial interpretation for the signed integers $a_i(n)$? There are many interpretations for $(-1)^{n-1} a_1(n)= (n-2)!+(n-3)!,$ see OEIS A001048. For $n\ge 4$ it is the size of the largest conjugacy class in $S_{n-1}.$ We were unable to find the other sequences $\{a_{i}(n)\}_{n\ge 3} $ in OEIS.
\end{ques}
\section{The subposet of normal words}
Let $N_{n,k}$ denote the poset of normal words of length at most $k$ in $A^*_{n,k},$ again with an artificial top element $\hat 1$ appended. Farmer showed that
\begin{theo}[Farmer~\cite{F}] $\mu(N_{n,k})=(-1)^{k-1} (n-1)^{k} =\mu(A^*_{n,k}),$ and $A_{n,k}, N_{n,k}$ both have the homology of a wedge of $(n-1)^{k}$ spheres of dimension $(k-1)$.
\end{theo}
Bj{\"o}rner and Wachs~\cite{BjWachs} showed that $N_{n,k}$ is dual CL-shellable and hence homotopy Cohen--Macaulay; it is therefore homotopy-equivalent to a wedge of $(n-1)^{k}$ spheres of dimension $(k-1)$.
The order complexes of the posets $A^*_{n,k}$ and $N_{n,k}$ are thus homotopy-equivalent.
Using Quillen's fibre theorem (\cite{Q}) we can establish a slightly stronger result:
\begin{lemma} Let $\alpha\in N^*_{n,k}.$ Then the intervals $(\hat 0,\alpha)_{N_{n,k}}$ and $(\hat 0,\alpha)_{A^*_{n,k}}$ are ${\stab(\alpha)}$-homotopy equivalent, for the stabiliser subgroup ${\stab(\alpha)}$ of $\alpha.$
In particular the homology groups are all ${\stab (\alpha)}$-isomorphic.
If $\alpha\in A^*_{n,k},$ but $\alpha\notin N^*_{n,k},$ then we know that the interval $(\hat 0, \alpha)_{ A^*_{n,k}}$ is contractible.
\end{lemma}
\begin{proof} Let $J_m$ be the set of words of length $m$ that are \emph{not} normal. Let
\[
B_j=(\hat 0, \alpha)_{A^*_{n,k}}\backslash (\cup_{m=j}^k J_m)
\]
be the subposet obtained by removing all normal words at rank $j$ and higher. Thus $B_1=(\hat 0, \alpha)_{N_{n,k}}.$ Set $B_{k+1}=(\hat 0, \alpha)_{A^*_{n,k}}.$ We claim that the inclusion maps
\begin{equation}
(\hat 0, \alpha)_{N_{n,k}}= B_1\subset B_2\subset \cdots \subset B_j\subset B_{j+1}\subset B_{k+1}=(\hat 0, \alpha)_{A^*_{n,k}}
\end{equation}
are group equivariant homotopy equivalences.
Note that
\[
B_j=B_{j+1}\backslash \{\text{non-normal words of length } j+1\},
\]
and $B_j$ coincides with $B_{j+1}$ for the first $j$ ranks. The fibres to be checked are $F_{\le w}=\{\beta\in B_j: \beta\le \alpha\},$ for $w\in B_{j+1}.$ If $w $ is a normal word in $B_{j+1},$ then $w\in B_j$ and the fibre is the half-closed interval $(\hat 0, w]$ in $B_j; $ it is therefore contractible. If $w\in B_{j+1}$ is not a normal word, then $w\notin B_j$ and the interval $(\hat 0, w)_{B_j}$ coincides with the same interval in $A^*_{n,k},$ so by Part~\eqref{prop7.4_1} of Proposition~\ref{LowerIntTopo}, it is contractible. Hence by Quillen's fibre theorem, the inclusion induces a homotopy equivalence.
\end{proof}
\begin{prop}\label{WHnormal} The Whitney homology modules of $A^*_{n,k}$ and $N_{n,k}$ are $S_n$-isomorphic.
In particular the $j$th Whitney homology of $N_{n,k}$ is isomorphic to the $S_n$-action on the elements at rank $j.$
\end{prop}
This statement is false for the dual Whitney homology. For instance,
\[
\mu(ab, abab)_{A^*_{n,k}}=+3,
\text{ but }
\mu(ab, abab)_{N_{n,k}}=+1.
\] The first interval is a rank-one poset with four elements $bab, aab, abb, aba,$ whereas the second consists only of two elements $aba, bab.$
It is also easy to find examples showing that the rank-selected homology is not the same for each poset.
\begin{exam}
Let $n=2$ and consider the rank-set $\{1,3\}$ for the poset $A^*_{2,k}$ and for its subposet of normal words $N_{2,k}$.
The words of length 3, all of which cover the two rank 1 elements $a$ and $b,$ are
\[
aaa, aab, aba, abb, baa, bab, bba, bbb.
\]
It is clear that the M{\"o}bius function values are
\[
\mu(\hat 0, aaa)=0=\mu(\hat 0, bbb), \mu(\hat 0, w)=-1 \text{ for all } w\notin \{aaa, bbb\}.
\]
Hence the M{\"o}bius number of the rank-selected subposet of all words is $-5,$ and from Theorem~\ref{rank-deletion} the $S_2$-representation on homology is $3\, S_{(2)}\oplus 2 \, S_{(1,1)}.$ The order complex is a wedge of 5 one-dimensional spheres.
Now consider the corresponding rank-selected subposet of normal words:
there are only two normal words of length 3, namely $aba, bab$ and hence the M{\"o}bius number of the rank-selected subposet of normal words is $-1,$ with trivial homology representation. The order complex is a one-dimensional sphere.
\end{exam}
\begin{exam}
More generally, let $S$ be the rank-set $[2,k]$, and consider the posets $A_{2,k}(S)$ and $N_{2,k}(S)$ obtained by deleting the atoms. Then by Theorem~\ref{rank-deletion} the homology of $A_{2,k}(S)$ is
$ (k-1)S_{(1^2)}^{\otimes k} +k S_{(2)},$
while the homology of the normal word subposet $N_{2,k}(S)$ is seen to be
$S_{(1^2)}^{\otimes k},$ which is either the trivial or the sign module, depending on the parity of $k.$
\end{exam}
\begin{rema}
In fact it is easy to see that $N_{2,k}$ is the ordinal sum~\cite{St3EC1} of $k$ copies of an antichain of size 2, with a bottom and top element attached. Hence for any subset $T$ of $[1,k]$, there is an $S_2$-equivariant poset isomorphism between $N_{2,k}(T)$ and $N_{2,|T|}.$ Since the $S_n$-homology of $N_{2,k}$ is easily seen to be the
$k$-fold tensor power of the sign representation,
this determines $\tilde{H}(N_{2,k}(T))$ for all rank-sets $T.$
%
Also, $S_2$ acts on the chains of any rank-selected subposet $N_{2,k}(T)$ like $2^{|T|-1}$ copies of the regular representation of $S_2,$ since the $2^k$ chains of $N_{2,k}$ break up into orbits of size 2.
\end{rema}
Recall from~\cite{St3EC1} that a finite graded poset $P$ with $\hat 0$ and $\hat 1$ is Eulerian if its M{\"o}bius function $\mu_P$ satisfies $\mu(P)=(-1)^{\rank (y)-\rank (x)}$ for all intervals $(x,y)\subseteq (\hat 0, \hat 1).$
It is known that all intervals $(x,y), y\neq \hat 1,$ in $N_{n,k}$ are Eulerian (see \eg \cite[Exercise~188]{St3EC1}). In fact Bj{\"o}rner and Wachs observed in~\cite{BjWachs} that for a finite alphabet $A=\{a_i: 1\le i\le n\},$ the poset of normal words without the top element
$N_{n,k}\backslash \{\hat 1\}$ is simply Bruhat order on the Coxeter group with $n$ generators $a_i$ and relations $a_i^2=1.$ Thus by lexicographic shellability~\cite{BjWachsAIM1982}, all intervals $(x,y), y\neq \hat 1$ are homotopy equivalent to a sphere.
In fact this makes $N_{n,k}\backslash \{\hat 1\}$ a $CW$-poset as defined in~\cite{Bj-CW}, and hence its order complex is isomorphic to the face poset of a regular $CW$-complex $\mathcal{K}$~\cite[Proposition~3.1]{Bj-CW}. It is easy to see from the definitions that if a finite group $G$ acts on a $CW$-poset $P$, then there is a $G$-module isomorphism between the $j$th Whitney homology $W\!H_j(P)$ and the $G$-action on the $(j-1)$-cells of the associated
$CW$-complex $\mathcal{K}(P)$, which are simply the elements of $P$ at rank $j$. This provides another explanation for the observation of Proposition~\ref{WHnormal}.
An EL-labelling of the dual poset of normal (or Smirnov) words appears in~\cite{TiansiLi}. In~\cite{LiSu} the program of the present paper is carried out for the subposet of Smirnov words.
\longthanks{The author is grateful to the anonymous reviewers for their detailed and valuable comments.}
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