text:
\section{Introduction}
\label{sec:intro}

Let $K$ be a finite simplicial complex with vertex set $V(K) \neq
\emptyset$ and let $r \geq 1$ and $s \geq 1$ be two natural numbers. A
map $\mathrm{col} \colon V(K) \to \{1,2,\ldots,r\}$ is an {\em
  $(r,s)$-coloring\/} of $K$ if there are no monochrome $s$-simplices
in $K$ \cite{DMN2010}.  We write $\chrps Krs$ for the number of
$(r,s)$-colorings of $K$.

\begin{defn}\label{defn:schpK}
  The {\em $s$-chromatic polynomial\/} of $K$ is the function $\chrps
  Krs$ of $r$.  The {\em $s$-chromatic number\/} of $K$, $\chs sK$, is
  the minimal $r \geq 1$ with $\chrps Krs>0$.
\end{defn}

The theorem below shows that $\chrps Krs$ is indeed polynomial in $r$
for fixed $K$ and $s$.  (By notational convention, $[r]_i = r(r-1)
\cdots (r-i+1)$ is the $i$th falling factorial in $r$.)

\begin{thm}\label{lemma:StK}
  The $s$-chromatic polynomial\ of $K$ is
  \begin{equation*}
    \chrps Krs = \sum_{i=\chs sK}^{|V(K)|} S(K,i,s)[r]_i
  \end{equation*}
  where $S(K,i,s)$ is the number of partitions of $V(K)$ into
  $i$ blocks containing no $s$-simplex of $K$.
\end{thm}

For $s=1$, an $(r,1)$-coloring of $K$ is a usual graph coloring,
$\chrps Kr1$ is the usual chromatic polynomial , and $\chs 1K$ the usual chromatic
number of the $1$-skeleton of $K$.  In general, $\chrps Krs$ depends
only on the $s$-skeleton of $K$.  Although the higher $s$-chromatic polynomial s for
simplicial complexes are analogues of $1$-chromatic polynomial s for graphs we shall
shortly see that there are structural differences between the cases
$s=$ and $s>1$.

Figure~\ref{fig:MBcol} shows a triangulation $\mathrm{MB}$ of the M\"obius\
band. To the left is a $(5,1)$- and to the right a $(2,2)$-coloring of
$\mathrm{MB}$. The chromatic polynomial s and chromatic numbers \footnote{The
  computations behind the examples of this note were carried out in
  the computer algebra system Magma \cite{magma}.} of $\mathrm{MB}$
are
\begin{equation*}
  \chrps {\mathrm{MB}}rs =
  \begin{cases}
     r^5 - 10 r^4 + 35 r^3 - 50 r^2 + 24 r & s= 1\\
     r^5 - 5 r^3 + 5 r^2 - r & s= 2\\
     r^5 & s \geq 3
  \end{cases} \qquad
  \chs s{\mathrm{MB}} =
  \begin{cases}
    5 & s=1 \\ 2 & s=2 \\ 1 & s \geq 3
  \end{cases}
\end{equation*}

\begin{figure}[h]
  \centering
\begin{tikzpicture}
  [vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  8pt, inner sep =0pt,draw}, vertex/.default=black, xscale=1.68, yscale=1.5]

\coordinate [vertex=red, label=above:$4$] (A1) at (-1,1);
\coordinate [vertex=red, label=above:$5$] (A2) at (0,1);
\coordinate [vertex=red, label=above:$1$] (A3) at (1,1);

\coordinate [vertex=red, label=below:$1$] (B1) at (-1.5,0);
\coordinate [vertex=blue, label=below:$2$] (B2) at (-.5,0);
\coordinate [vertex=blue, label=below:$3$] (B3) at (.5,0);
\coordinate [vertex=red, label=below:$4$] (B4) at (1.5,0);

\draw (B1) -- node[sloped] {$>$} (A1) -- (A2) -- (A3) -- node[sloped]
{$>$} (B4);
\draw (B1) -- (B2) -- (B3) -- (B4);
\draw (A1) -- (B2) -- (A2) -- (B3) -- (A3);

\begin{scope}[xshift=-4cm]

\coordinate [vertex=red, label=above:$4$] (A1) at (-1,1);
\coordinate [vertex=magenta, label=above:$5$] (A2) at (0,1);
\coordinate [vertex=blue, label=above:$1$] (A3) at (1,1);

\coordinate [vertex=blue, label=below:$1$] (B1) at (-1.5,0);
\coordinate [vertex=yellow, label=below:$2$] (B2) at (-.5,0);
\coordinate [vertex=green, label=below:$3$] (B3) at (.5,0);
\coordinate [vertex=red, label=below:$4$] (B4) at (1.5,0);

\draw (B1) -- node[sloped] {$>$} (A1) -- (A2) -- (A3) -- node[sloped]
{$>$} (B4);
\draw (B1) -- (B2) -- (B3) -- (B4);
\draw (A1) -- (B2) -- (A2) -- (B3) -- (A3);
\end{scope}

\end{tikzpicture}

\caption{A $(5,1)$-coloring and a $(2,2)$-coloring of a $5$-vertex
  triangulated M\"obius band $\mathrm{MB}$}
  \label{fig:MBcol}
\end{figure}

\subsection{Notation}
\label{sec:notation}
We shall use the following notation throughout the paper:
\begin{description}
\item[$K$] a finite simplicial complex
\item[$K^s$]  the $s$-skeleton of $K$
\item[$F^s(K)$]  the set of $s$-simplices $K$

\item[$\# V$ or $|V|$] the number of elements in the finite set $V$

\item[$V(K)$] the vertex set $\bigcup K$ of $K$ and $m(K)=|V(K)$ is
  the number of vertices in $K$

\item[{$D[V]$}] the complete simplicial complex of {\em all\/} subsets
  of the finite set $V$
\item[{$[m]$}] the finite set $\{1,\ldots,m\}$ of cardinality $m$
\item[{$[r]_i$}] the $i$the falling factorial polynomial $[r]_i = i!
  \binom ri$ in $r$
\item[$P(a,b)$] the open interval $(a,b)$ in the poset $P$
\end{description}

\section{Three ways to the $s$-chromatic polynomial\ of a simplicial complex}
\label{sec:schp}

In this section we present three different to approaches to the
$s$-chromatic polynomial\ $\chrps Krs$:
\begin{itemize}
\item Theorem~\ref{thm:connindep} via $1$-chromatic polynomial s of graphs;
\item Theorem~\ref{thm:LsK}
via the M\"obius\ function for the $s$-chromatic lattice;
\item Theorem~\ref{lemma:StK} via the simplicial $s$-Stirling numbers
  of the second kind.
\end{itemize}

\subsection{Block-connected $s$-independent vertex partitions}
\label{sec:conpartV}

Let $s \geq 1$ be a natural number.

\begin{defn}\label{defn:independent}
Let $B \subset V(K)$ be a set of vertices of $K$. Then
\begin{itemize}
\item $B$ is {\em $s$-independent\/} if $B$ contains no $s$-simplex of
  $K$;
\item $B$ is  {\em connected\/} if $K \cap D[B]$ is a
  connected simplicial complex;
\item the {\em connected components\/} of $B$ are the maximal
  connected subsets of $B$.
 \end{itemize}
\end{defn}

\begin{defn}\label{defn:cP}
  Let $P$ be a partition of $V(K)$.
  \begin{itemize}
  \item The {\em   graph\/} $G_0(P)$ of $P$ is the simple graph
    whose vertices are the blocks of $P$ and with two blocks connected
    by and edge if their union is connected;
  \item The {\em block-connected refinement\/} $P_0$ of $P$ is the
    refinement whose blocks are the connected components of the blocks
    of $P$;
  \item $P$ is {\em block-connected\/} if the blocks of $P$ are
    connected (ie if $P=P_0$).
  \end{itemize}
\end{defn}

\begin{lemma}\label{lemma:diffblocks}
  Let $P$ be a partition of $V(K)$. If two different blocks of the
  block-connected refinement $P_0$ are connected by an edge in the
    graph $G_0(P_0)$ of $P$ then they lie in different blocks of
  $P$.
\end{lemma}
\begin{proof}
  The connected components of the blocks of $P$ are maximal with
  respect to connectedness.
\end{proof}

\begin{defn}
  $\mathrm{BCP}^s(K)$ is the set of all block-connected $s$-independent
  partitions of $V(K)$.
\end{defn}

Recall that $\chrps {G_0(P)}r1$ is the $1$-chromatic polynomial\ of the simple graph
$G_0(P)$ of the partition $P$.

\begin{thm}\label{thm:connindep}
  The $s$-chromatic polynomial\ for $K$ is the sum
  \begin{equation*}
    \chrps Krs = \sum_{P \in \mathrm{BCP}^s(K)} \chrps {G_0(P)}r1
  \end{equation*}
  of the $1$-chromatic polynomials and
 the $s$-chromatic number of $K$ is the minimum
\begin{equation*}
    \chs sK = \min_{P \in \mathrm{BCP}^s(K)} \chs 1{G_0(P)}
\end{equation*}
of the $1$-chromatic numbers for the   graphs of all the
block-connected $s$-independent partitions of $V(K)$.
\end{thm}
\begin{proof}
  Let $\mathrm{col} \colon V(K) \to [r] $ be an $(r,s)$-coloring of
  $K$.  The monochrome partition $P(\mathrm{col})$ of $V(K)$ is the
  $s$-independent partition whose blocks are the nonempty monochrome
  sets of vertices $\{\mathrm{col}=i\}$ for $i \in [r]$.  The
  block-connected refinement $P(\mathrm{col})_0$ of the monochrome
  partition is a block-connected $s$-independent partition of $K$. The
  original coloring $\mathrm{col}$ of $K$ is also a coloring of the
  graph $G_0(P(\mathrm{col})_0)$ of $P(\mathrm{col})_0$ for, by
  Lemma~\ref{lemma:diffblocks}, distinct vertices of $1$-simplices of
  this graph have distinct colors. We have shown that any
  $(r,s)$-coloring $\mathrm{col}$ of $K$ induces an $(r,1)$-coloring
  $\mathrm{col}_0$ of the graph $G_0(P(\mathrm{col})_0)$ of the
  block-connected refinement of the monochrome partition.

  Let $P \in \mathrm{BCP}^s(K)$ be a block-connected $s$-independent
  partition of $V(K)$ and $\mathrm{col}_0 \colon P \to \{1,\ldots,r\}$
  an $(r,1)$-coloring of its graph $G_0(P)$.  Then $\mathrm{col}_0$
  determines a map $\mathrm{col} \colon V(K) \to [r]$ that is constant
  on the blocks of $P$.  An $s$-simplex of $K$ can not be monochrome
  under $\mathrm{col}$ as it intersects at least two different blocks
  of $P$ connected by an edge of $G_0(P)$. Thus $\mathrm{col}$ is an
  $(r,s)$-coloring of $K$.

  These two constructions are inverses of each other.
\end{proof}

\begin{rmk}[The minimal block-connected $s$-independent partition]
 \label{rmk:C0}
Let $C_0=\{\{v\} \mid v \in V(K)\}$ be the block-connected
$s$-independent partition of $V(K)$ whose blocks are singletons. The
  graph $G_0(C_0) = K^1$ is the $1$-skeleton of $K$.  Thus
the $1$-chromatic polynomial\ of the $1$-skeleton of $K$ is always one of the
polynomials in the sum of Theorem~\ref{thm:connindep}. If $K$ is
$1$-dimensional, $\mathrm{BCP}^1(K)$ consists only of the partition
$C_0$ and Theorem~\ref{thm:connindep} simply says that the $1$-chromatic polynomial\
of a simplicial complex is the $1$-chromatic polynomial\ of its $1$-skeleton.
\end{rmk}

\begin{exmp}[The block-connected $2$-independent partitions for {$D[3]$}]
  The $2$-simplex $D[3]$ has $4$ block-connected $2$-independent
  partitions $C_0$, $\{\{1\},\{2,3\}\}$, $\{\{2\},\{1,3\}\}$, and
  $\{\{3\},\{1,3\}\}$. The   graph of $C_0$ is the
  complete graph $K_3$, the $1$-skeleton of $D[3]$. The graphs of the
  other three partitions are all the complete graph $K_2$. Thus the
  $2$-chromatic polynomial\ of $D[3]$ is $\chrps {D[3]}r2 = \chrps {K_3}r1 + 3\chrps
  {K_2}r1 = [r]_3 + 3[r]_2 = [r]_2(r+1) = r^3-r$ and the $2$-chromatic
  number is $\chs 2{D[3]} = 2$.
\end{exmp}

\begin{exmp}[A $(2,2)$-coloring and the graph of the block-connected
  refinement of its monochrome partition]\label{rmk:T2}
  The picture below illustrates a $(2,2)$-coloring of a $9$-vertex
  triangulation of the torus
  \begin{center}
 \begin{tikzpicture}
  [vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  8pt, inner sep =0pt,draw}, vertex/.default=black]

\begin{scope}[scale=1.5]
  \draw [step = 1cm] (1,1) grid (4,4);
  \draw (1,1) -- (4,4);
  \draw (3,1) -- (4,2);
  \draw (2,1) -- (4,3);
  \draw (1,2) -- (3,4);
  \draw (1,3) -- (2,4);

  \node (a) at (1,4) [vertex=red,label=above left:$1$] {};
  \node (b) at (2,4) [vertex=blue,label=above:$2$] {};
  \node (c) at (3,4) [vertex=blue,label=above:$3$] {};
  \node (d) at (4,4) [vertex=red,label=above right:$1$] {};

  \node  at (1,3) [vertex=blue,label=left:$5$] {};
  \node  at (2,3) [vertex=red,label=below right:$8$] {};
  \node  at (3,3) [vertex=blue,label=below right:$9$] {};
  \node  at (4,3) [vertex=blue,label=right:$5$] {};

  \node  at (1,2) [vertex=blue,label=left:$4$] {};
  \node  at (2,2) [vertex=blue,label=below right:$6$] {};
  \node  at (3,2) [vertex=red,label=below right:$7$] {};
  \node  at (4,2) [vertex=blue,label=right:$4$] {};

  \node  at (1,1) [vertex=red,label=below left:$1$] {};
  \node  at (2,1) [vertex=blue,label=below:$2$] {};
  \node  at (3,1) [vertex=blue,label=below:$3$] {};
  \node  at (4,1) [vertex=red,label=below right:$1$] {};
\end{scope}
  \begin{scope}[xshift=7cm,yshift=1.25cm]
    \node (C1) at (3.5,2.5)
    [rectangle,draw]  {\phantom{\rule[0pt]{160pt}{35pt}}};
     \node (C2) at (3.5,0.5)
    [circle,draw]  {\phantom{\rule[0pt]{15pt}{15pt}}};
    \node (C3) at (1,4.5)
    [circle,draw]  {\phantom{\rule[0pt]{15pt}{15pt}}};
     \node (C4) at (6,4.5)
    [circle,draw]  {\phantom{\rule[0pt]{15pt}{15pt}}};

    \node at (1,2.4) [vertex=blue, label= above:$2$] {};
    \node at (2,2.4) [vertex=blue, label= above:$3$] {};
    \node at (3,2.4) [vertex=blue, label= above:$4$] {};
    \node at (4,2.4) [vertex=blue, label= above:$5$] {};
    \node at (5,2.4) [vertex=blue, label= above:$6$] {};
    \node at (6,2.4) [vertex=blue, label= above:$9$] {};

    \node at (3.5,0.4) [vertex=red, label= above:$1$] {};
    \node at (1,4.4) [vertex=red, label= above:$7$] {};
    \node at (6,4.4) [vertex=red, label= above:$8$] {};

    \draw (C1) -- (C2);
    \draw (C1) -- (C3);
    \draw (C1) -- (C4);
  \end{scope}
\end{tikzpicture}
\end{center}
and its corresponding graph. There are $6937$ block-connected
partitions of the vertex set, and $3$ of them has the graph shown
above. The $2$-chromatic polynomial is $21[r]_2+ 742[r]_3 + 3747[r]_4
+ 4908[r]_5 + 2295[r]_6 + 444[r]_7 + 36[r]_8 + [r]_9 = [r]_2(r^7 + r^6
- 17r^5 + 10r^4 + 82r^3 - 116r^2 - 23r + 67)$ and the $2$-chromatic
number is $2$.
\end{exmp}

\begin{exmp}[The $(r,2)$-colorings of a simplicial complex $K$]\label{exmp:K}
  Let $K$ be the pure $2$-dimensional complex with facets $F^2(K) =
  \{\{1,2,3\},\{2,3,4\},\{4,5,6\}\}$.
  \begin{center}
    \begin{tikzpicture}
[vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  8pt, inner sep =0pt,draw}, vertex/.default=black,scale=0.75,xscale=1.5]

  \path [fill = green!20] (0,0) -- (2,2) -- (2,-2) -- (0,0);
  \path [fill = green!20] (4,0) -- (2,2) -- (2,-2) -- (4,0);
  \path [fill = green!20] (4,0) -- (6,2) -- (6,-2) -- (4,0);

  \draw (2,2) -- (2,-2);
  \draw (0,0) -- (2,2) -- (4,0) -- (6,2) -- (6,-2) -- (4,0) -- (2,-2)
  -- (0,0) -- cycle;

  \draw (0,0) node[vertex=red, label=left:$1$] {};
  \draw (2,2) node[vertex=red, label=above:$2$] {};
  \draw (2,-2) node[vertex=blue, label=below:$3$] {};
  \draw (4,0) node[vertex=blue, label=below:$4$] {};
  \draw (6,-2) node[vertex=red, label=below:$5$] {};
  \draw (6,2) node[vertex=red, label=above:$6$] {};
\end{tikzpicture} \hspace{1cm}
 \begin{tikzpicture}
[vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  8pt, inner sep =0pt,draw},
vertex/.default=black,scale=0.75]

 \draw (0,0) node[vertex=red] {};
  \draw (3,2) node[vertex=red] {};
  \draw (3,-2) node[vertex=blue] {};
  \draw (6,0) node[vertex=blue] {};
  \draw (9,-2) node[vertex=red] {};
  \draw (9,2) node[vertex=red] {};

  \path (1.5,1) node(a) [ellipse, rotate=-57, draw]
 {\phantom{\rule[0pt]{20pt}{60pt}}};

 \path (4.5,-1) node(b) [ellipse, rotate=-57, draw]
 {\phantom{\rule[0pt]{20pt}{60pt}}};

 \path (9,0) node(c) [ellipse, rotate=0, draw]
 {\phantom{\rule[0pt]{20pt}{70pt}}};

 \draw (a) -- (b) -- (c);

\end{tikzpicture} \hspace{2cm}
\end{center}
The picture shows a $(2,2)$-coloring of $K$ and the corresponding
$(2,1)$-coloring of the associated graph, $G_0(P_0)$, the block
connected refinement of the monochrome partition
$P=\{\{1,2,5,6\},\{3,4\}\}$.  Table~\ref{tab:G0PforK} shows the graphs
$G_0(P)$ for all block connected partitions $P \in \mathrm{BCP}^2(K)$.
For each graph, the table records its $1$-chromatic polynomial and its
$1$-chromatic number. The $2$-chromatic polynomial of $K$ is $\chrps
K22 = 15[r]_2 + 73[r]_3 + 62[r]_4 + 15[r]_5 + [r]_6 =
[r]_2(r-1)(r+1)(r^2+r-1)$ and the $2$-chromatic number is $\chs 2K =
2$.

\begin{table}[t]
  \centering
\begin{tabular}[t]{c|>{\centering\arraybackslash}m{2.2cm}|c|c}
    $\#$  in $\mathrm{BCP}^2(K)$ & $G_0(P)$ & $\chrps {G_0(P)}r1$
    & $\chs 1{G_0(P)}$ \\ \hline
    $1$ & \vspace{2mm}
    \begin{tikzpicture}
      [vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  3pt, inner sep =0pt,draw}, vertex/.default=black,scale=.25,xscale=1.3]
    \draw (0,0) node[vertex] {} --
    (2,2) node[vertex] {} --
   (4,0) node[vertex] {} --
   (6,2) node[vertex] {} --
   (6,-2) node[vertex] {} --
   (4,0) node[vertex] {} --
    (2,-2) node[vertex] {} --
    (0,0) -- cycle;
    \draw (2,2) -- (2,-2);
    \end{tikzpicture} &
    $r(r-1)^2(r-2)^3$ &  $3$  \\

    $1$ &
     \begin{tikzpicture}
      [vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  3pt, inner sep =0pt,draw}, vertex/.default=black,scale=.25,xscale=1.3]
    \draw (0,2) node[vertex] {} --
    (0,-2) node[vertex] {} --
   (2,0) node[vertex] {} --
   (4,0) node[vertex] {} --
   (6,0) node[vertex] {};
    \draw (0,2) -- (2,0);
    \end{tikzpicture} &
    $r(r-1)^3(r-2)$ & $3$ \\
    $3$ &
      \begin{tikzpicture}
      [vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  3pt, inner sep =0pt,draw}, vertex/.default=black,scale=.25,xscale=1.3]
    \draw (0,0) node[vertex] {} --
    (2,-2) node[vertex] {} --
   (4,0) node[vertex] {} --
   (6,0) node[vertex] {};
   \draw (0,0) --
   (2,2) node[vertex] {} -- (4,0);
   \draw (2,2) -- (2,-2);
    \end{tikzpicture} &
    $r(r-1)^2(r-2)^2$ & $3$ \\
    $4$ &
      \begin{tikzpicture}
      [vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  3pt, inner sep =0pt,draw}, vertex/.default=black,scale=.25,xscale=1.3]
    \draw (0,2) node[vertex] {} --
    (0,-2) node[vertex] {} --
   (2,0) node[vertex] {} --
   (4,2) node[vertex] {} --
   (4,-2) node[vertex] {} -- (2,0);
   \draw (0,2) -- (2,0);
    \end{tikzpicture} &
    $r(r-1)^2(r-2)^2$ & $3$ \\
    $16$ &
     \begin{tikzpicture}
      [vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  3pt, inner sep =0pt,draw}, vertex/.default=black,scale=.25,xscale=1.3]
    \draw (0,2) node[vertex] {} --
    (0,-2) node[vertex] {} --
   (2,0) node[vertex] {} --
   (4,0) node[vertex] {};
   \draw (0,2) -- (2,0);
    \end{tikzpicture} &
    $r(r-1)^2(r-2)$  & $3$ \\
    $3$ &
      \begin{tikzpicture}
      [vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  3pt, inner sep =0pt,draw}, vertex/.default=black,scale=.25,xscale=1.3]
    \draw (0,0) node[vertex] {} --
    (2,0) node[vertex] {} --
   (4,0) node[vertex] {} --
   (6,0) node[vertex] {};
    \end{tikzpicture} &
   $r(r-1)^3$ &  $2$ \\
     $12$ &
      \begin{tikzpicture}
      [vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  3pt, inner sep =0pt,draw}, vertex/.default=black,scale=.25,xscale=1.3]
    \draw (0,0) node[vertex] {} --
    (2,0) node[vertex] {} --
   (4,0) node[vertex] {};
    \end{tikzpicture} &
    $r(r-1)^2$ & $2$

  \end{tabular}
  \caption{The graphs for the block-connected partitions in
           $\mathrm{BCP}^2(K)$}
  \label{tab:G0PforK}
\end{table}
\end{exmp}

  \begin{exmp}[The $(r,2)$-colorings of the M\"obius\ band]
    The set $\mathrm{BCP}^2(\mathrm{MB})$ of block-connected
    $2$-independent partitions of the triangulated M\"obius\ band
    $\mathrm{MB}$ (Figure~\ref{fig:MBcol}) has $36$ elements. There
    are $5,5,15,10,1$ partitions in $\mathrm{BCP}^2(\mathrm{MB})$
    realizing the partitions $[3,2]$, $[3,1,1]$, $[2,2,1]$,
    $[2,1,1,1]$, $[1,1,1,1,1]$ of the integer $|V(\mathrm{MB})|=5$.
    All associated graphs are complete graphs.

    This yields the $2$-chromatic polynomial\ $\chrps {\mathrm{MB}}r2 = 5[r]_2 +
    20[r]_3 + 10[r]_4 + [r]_5 = [r]_2(r^3 + r^2 - 4r + 1) = r^5 - 5r^3
    + 5r^2 - r$ and the $2$-chromatic number is $\chs 2{\mathrm{MB}} =
    2$.
\end{exmp}

\begin{rmk}[The $\mathcal{S}$-chromatic polynomial of
  $K$]\label{rmk:Schrpol}
  Let $\mathcal{S}$ be a set of connected subcomplexes of $K$. A set
  $B \subset V(K)$ of vertices is $\mathcal{S}$-independent if $B$ is
  not a superset of any member of $\mathcal{S}$. Let
  $\mathrm{BCP}^\mathcal{S}(K)$ be the set of
  $\mathcal{S}$-independent partitions of $V(K)$. An
  $(r,\mathcal{S})$-coloring is a map $V(K) \to \{1,\ldots,r\}$ such
  that $\#\mathrm{col}(S)>1$ for all $S \in \mathcal{S}$. The number
  of $(r,\mathcal{S})$-colorings of $K$ is
 \begin{equation*}
    \chrps Kr{\mathcal{S}} =
    \sum_{P \in \mathrm{BCP}^\mathcal{S}(K)} \chrps {G_0(P)}r1
  \end{equation*}
  as one sees by an obvious generalization of
  Theorem~\ref{thm:connindep}. An $(r,s)$-coloring of $K$ is an
  $(r,\mathcal{S})$-coloring of $K$ where $\mathcal{S} = F^s(K)$ is
  the set of $s$-simplices.
\end{rmk}

\subsection{The $s$-chromatic linear program}
\label{sec:linprg}
Read \cite[\S10]{read68} explains how to construct a linear program
with minimal value equal to the $s$-chromatic number $\chs sK$ of $K$.

\marginpar{\textcolor{red}{maximal?}}

\begin{defn}
  $M^s(K)$ is the set of all maximal $s$-independent subsets of
  $V(K)$.
\end{defn}

Let $A$ be the $(m(K) \times |M^s(K)|)$-matrix
\begin{equation*}
  A(v,M) =
  \begin{cases}
    1 & v \in M \\
    0 & v \not\in M
  \end{cases}
\end{equation*}
recording which vertices $v \in V(K)$ belong to which maximal $s$-independent
sets $M \in M^s(K)$. Now the $s$-chromatic number
\begin{equation*}
  \chs sK = \min \{ \sum_{M \in M^s(K)} x(M) \mid
  x \colon M^s(K) \to \{0,1\}, \forall v \in V(K) \colon \sum_{M \in M^s(K)}
  A(v,M)x(M) \geq 1 \}
\end{equation*}
is the minimal value of the objective function $\sum_{M \in M^s(K)} x(M)$
in  $|M^s(K)|$
variables $x \colon M^s(K) \to \{0,1\}$, taking values $0$ or $1$, and
$m(K)$ constraints $\sum_{M \in M^s(K)} A(v,M)x(M) \geq 1$, $v \in V(K)$.

\subsection{The $s$-chromatic lattice}
\label{sec:chrlattice}

Our approach here simply follows Rota's classical method for computing
chromatic polynomial s from M\"obius\ functions of lattices \cite[\S9]{rota64}.  We need
some terminology in order to characterize the monochrome loci for
colorings of $K$. Recall that $F^s(K)$ is the set of $s$-simplices of
$K$.

\begin{defn}
  Let $S \subset F^s(K)$ be a set of $s$-simplices of $K$.
  \begin{itemize}
  \item The equivalence relation $\sim$  is the smallest equivalence
    relation in $S$
     such that $s_1 \cap
    s_2 \neq \emptyset \Longrightarrow s_1 \sim s_2$ for all $s_1, s_2
    \in S$;
  \item the connected components of $S$ are the equivalence classes
    under $\sim$;
  \item $\pi_0(S)$ is the set of connected components of $S$;
  \item $S$ is {\em connected\/} if it has at most one component;
  \item $V(S) = \bigcup S$ is the {\em vertex set\/} of $S$
  \item $\pi(S)$ is the partition of $V(K)$ whose blocks are the
    vertex sets of the connected components of $S$ together with the
    singleton blocks $\{v\}$, $v \in V(K)-V(S)$, of vertices not in
    any simplex in $S$;
  \item $S$ is {\em closed\/} if $S$ contains any $s$-simplex in $K$
    contained in the vertex set of $S$, ie if
\begin{equation*}
  \{ \sigma \in F^s(K) \mid \sigma \subset V(S) \} = S
\end{equation*}
\item the {\em closure\/} of $S$ is the smallest closed set of $s$-simplices
containing $S$.
  \end{itemize}
\end{defn}

For instance, the empty set $S=\emptyset$ of $0$ $s$-simplices is
connected with $0$ connected components. If $K=D[4]$, the set
$\{\{1,2\},\{2,4\}\}$ of $1$-simplices is connected while
$\{\{1,2\},\{3,4\}\}$ has the two components $\{\{1,2\}\}$ and
$\{\{3,4\}\}$.

A set of $s$-simplices is closed if and only if it equals its closure.
For instance in $F^2(D[5])$, the set $\{\{1,2,3\},\{3,4,5\}\}$ is not
closed because its closure is the set of all $2$-simplices in $D[5]$.
The empty set of $s$-simplices, any set of just one $s$-simplex, and
any set of disjoint $s$-simplices are closed.

In this picture the green set of $2$-simplices is
\begin{center}
 \begin{tikzpicture}
  [vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  4pt, inner sep =0pt,draw}, vertex/.default=black]

\begin{scope}[yscale=1]
 \path [fill=green!20] (0,0) -- (0,1) -- (1.12,0) --(0,0);
 \path [fill=green!20] (0,1) -- (0,2) -- (1.12,2) --(0,1);
 \path [fill=green!20] (1.12,2) -- (2.24,2) -- (2.24,1) --(1.12,2);
 \path [fill=green!20] (2.24,1) -- (2.24,0) -- (1.12,0) --(2.24,1);

  \node (a) at (0,0) [vertex] {};
  \node (b) at (0,1) [vertex] {};
  \node (c) at (0,2) [vertex] {};
  \node (d) at (1.12,2) [vertex] {};
  \node (e) at (2.24,2) [vertex] {};
  \node (f) at (2.24,1) [vertex] {};
  \node (g) at (2.24,0) [vertex] {};
  \node (h) at (1.12,0) [vertex] {};

  \draw (a) -- (c) -- (e) -- (g) -- (a) -- cycle;
  \draw (b) -- (d) -- (f);
  \draw (b) -- (h) -- (f);
  \draw (d) -- (h);
\end{scope}

\begin{scope}[yscale=1,xshift=4cm]

 \path [fill=green!20] (0,1) -- (0,2) -- (1.12,2) --(0,1);

 \path [fill=green!20] (2.24,1) -- (2.24,0) -- (1.12,0) --(2.24,1);

  \node (a) at (0,0) [vertex] {};
  \node (b) at (0,1) [vertex] {};
  \node (c) at (0,2) [vertex] {};
  \node (d) at (1.12,2) [vertex] {};
  \node (e) at (2.24,2) [vertex] {};
  \node (f) at (2.24,1) [vertex] {};
  \node (g) at (2.24,0) [vertex] {};
  \node (h) at (1.12,0) [vertex] {};

  \draw (a) -- (c) -- (e) -- (g) -- (a) -- cycle;
  \draw (b) -- (d) -- (f);
  \draw (b) -- (h) -- (f);
  \draw (d) -- (h);
\end{scope}

\begin{scope}[yscale=1,xshift=8cm]

 \path [fill=green!20] (0,1) -- (1.12,2) -- (1.12,0) --(0,1);
 \path [fill=green!20] (1.12,2) -- (1.12,0) -- (2.24,1) --(1.12,2);

  \node (a) at (0,0) [vertex] {};
  \node (b) at (0,1) [vertex] {};
  \node (c) at (0,2) [vertex] {};
  \node (d) at (1.12,2) [vertex] {};
  \node (e) at (2.24,2) [vertex] {};
  \node (f) at (2.24,1) [vertex] {};
  \node (g) at (2.24,0) [vertex] {};
  \node (h) at (1.12,0) [vertex] {};

  \draw (a) -- (c) -- (e) -- (g) -- (a) -- cycle;
  \draw (b) -- (d) -- (f);
  \draw (b) -- (h) -- (f);
  \draw (d) -- (h);
\end{scope}
\end{tikzpicture}
\end{center}
connected and not closed, closed and not connected, closed and
connected, respectively.

The partition $\pi(S)$ has $|\pi(S)| =
|\pi_0(S)|+m(K)-|V(S)|$ blocks.

\begin{lemma}\label{lemma:closedcomp}
  Let $S$ be a set of $s$-simplices in $K$ and $S_0$ a connected
  component of $S$. Then $S_0$ is closed if and only if
  \begin{equation*}
    \{ \sigma \in F^s(K) \mid \sigma \subset V(S_0) \} \subset S
  \end{equation*}
\end{lemma}
\begin{proof}
  Since the condition is certainly necessary we only need to see that
  it is sufficient. Let $\sigma$ be an $s$-simplex in $K$ with all its
  vertices in $V(S_0)$. Then $\sigma$ lies in $S$ by assumption. But
  $\sigma$ is equivalent to all elements of the equivalence class
  $S_0$. Thus $\sigma \in S_0$.
\end{proof}

\begin{lemma}\label{lemma:meet}
  Let $S$ and $T$ be sets of $s$-simplices in $K$.
  \begin{enumerate}
  \item \label{lemma:meet1} If $S$ and $T$ are closed, so is $S \cap T$.
  \item \label{lemma:meet2} If $S$ and $T$ have closed connected
    components, so does $S \cap T$
  \end{enumerate}
\end{lemma}
\begin{proof}

  \noindent \eqref{lemma:meet1}
  Let $\sigma$ be an $s$-simplex of $K$ and suppose that $\sigma
  \subset V(S \cap T)$. Then $\sigma \subset V(S)$ an $\sigma \subset
  V(T)$ so that $\sigma \in S$ and $\sigma \in T$ as $S$ and $T$ are
  closed.

  \noindent \eqref{lemma:meet2} Let $R$ be a connected component of $S
  \cap T$. Let $S_0$ be the connected component of $S$ containing $R$
  and $T_0$ be the connected component of $T$ containing $R$.  Then $R
  \subset S_0 \cap T_0$.  Suppose that $\sigma \in F^s(K)$ is an
  $s$-simplex with $\sigma \subset V(R)$. Then $\sigma \subset V(S_0
  \cap T_0)$ so $\sigma \in S_0 \cap T_0$ by \eqref{lemma:meet1} as
  the connected components $S_0$ and $T_0$ are assumed to be closed.
  In particular, $\sigma \in S \cap T$. According to
  Lemma~\ref{lemma:closedcomp}, the connected component $R$ is closed.
\end{proof}

\begin{defn}\label{defn:LsK}
  The $s$-chromatic lattice of $K$ is the set $L^s(K)$ of all subsets
  of $F^s(K)$ with closed connected components.  $L^s(K)$ is a
  partially ordered by set inclusion.
\end{defn}

The set $L^s(K)$ contains the empty set $\emptyset$ of $s$-simplices
and the set $F^s(K)$ of all $s$-simplices. These two elements of
$L^s(K)$ are distinct when $K$ has dimension at least $s$.

\begin{cor}
  $L^s(K)$ is a finite lattice with $\widehat 0 = \emptyset$,
  $\widehat 1 = F^s(K)$, and meet $S \wedge T = S \cap T$.
\end{cor}
\begin{proof}
  If $S,T \in L^s(K)$ then $S \cap T$ is also in $L^s(K)$ by
  Lemma~\ref{lemma:meet} and this is clearly the greatest lower
  bound of $S$ and $T$. It is now a standard result that $L^s(K)$ is a
  finite lattice \cite[Proposition 3.3.1]{stanley97}. The join $S \vee
  T$ of $S,T \in L^s(K)$ is the intersection of all supersets $U \in
  L^s(K)$ of $S \cup T$.
\end{proof}

\begin{exmp}[The $s$-chromatic lattice {$L^s(D[m])$}]
  \label{exmp:L2Dn+} The closed and connected elements of the
  $s$-chromatic lattice $L^s(D[m])$ of the complete simplex $D[m]$ on
  $m>s$ vertices are $\emptyset$ and the $\binom mk$ sets $F^s(D[k])$
  of all $s$-simplices in the subcomplexes $D[k]$ for $s < k \leq m$.
  The map $S \to \pi(S)$ is an isomorphism\ between the lattice $L^s(D[m])$
  and the lattice, ordered by refinement, of all partitions of the set
  $[m]$ into blocks of size $>s$ or $1$.  The least element, $\widehat
  0 = (1) \cdots (m)$, is the partition with $m$ blocks and the
  greatest element, $\widehat 1 = (1\cdots m)$, the partition with $1$
  block.  $L^s(D[m])$ is not a graded lattice \cite[p 99]{stanley97}
  in general when $s \geq 2$. To see this, observe that the
  $2$-chromatic lattices $L^2(D[3])$, $L^2(D[4])$, and $L^2(D[4])$ are
  graded but the lattice $L^2(D[6])$ is not graded as it contains two
  maximal chains
  \begin{gather*}
    \widehat 0 = (1)(2)(3)(4)(5)(6) < (123)(4)(5)(6) <
    (1234)(5)(6) < (12345)(6) < (123456) = \widehat 1 \\
     \widehat 0 = (1)(2)(3)(4)(5)(6) < (123)(4)(5)(6) < (123)(456) <
    (123456) = \widehat 1
  \end{gather*}
  of unequal length. In contrast, the $1$-chromatic lattice of any
  finite simplicial complex is always graded and even geometric
  \cite[\S9, Lemma~1]{rota64}.
\end{exmp}

  \begin{rmk}[The M\"obius\ function for the $s$-chromatic lattices
    {$L^s(D[m])$}] \label{rmk:wlattice} Our discussion of the M\"obius\
    function for the lattice $L^s(D[m])$ echoes the exposition of the
    M\"obius\ function for the geometric lattice $L^1(D[m])$ of {\em all\/}
    partitions from \cite[Example 3.10.4]{stanley97}.

  Let $w \colon [m] \to \mathbf{N}$ be a function that to every element of
  $[m]$ associates a natural number, thought of as a weight function.
  We write $w=1^{i_1}2^{i_2}\cdots r^{i_r}$, or something similar, for
  the weight function $w$ defined on the set $[m]$ of cardinality $m =
  \sum_j i_j$ and mapping $i_j$ elements to $j$ for $1 \leq
  j \leq r$. The map $w$ extends to a map, also called $w$, defined on
  the set of all nonempty subsets $X$ of $[m]$ given by $w(X) =
  \sum_{x \in X}w(x)$.  Let $L^s_m(w)$ be the lattice of all
  partitions of the set $[m]$ into blocks $X$ that are singletons or
  have weight $w(X)>s$. The non-singleton blocks of the meet $\sigma
  \wedge \tau$ of two partitions $\sigma, \tau \in L^s_m(w)$ are the
  subsets of weight $>s$ of the form $S \cap T$ where $S$ is a block
  in $\sigma$ and $T$ a block in $\tau$.  Write $\mu_m^s(w)$ for the
  M\"obius\ function of $L^s_m(w)$.

  In particular, $L^s_m(1^m)$ is a synonym for $L^s(D[m])$ and we are
  primarily interested in the M\"obius\ function $\mu_m^s(1^m)$ of the
  uniform weight $w=1^m$. However, the computation of this M\"obius\
  function will involve the M\"obius\ functions of other weights as
  well. We shall therefore discuss the M\"obius\ functions  $\mu_m^s(w)$
  for general weight functions $w$.

  Suppose that $\sigma \in L^s_m(w)$, $\sigma < \widehat 1$, is a
  partition of $[m]$ into singleton blocks or blocks of weight $>s$.
  Let $w(\sigma)$ be the restriction of $w$ to the set of blocks of
  $\sigma$. Thus $w(\sigma)(X) = \sum_{x \in X}w(x)$ for any block $X$
  of $\sigma$. Then the interval
  \begin{equation*}
   L^s_m(w) \supset [\sigma, \widehat 1] = L^s_{|\sigma|} (w(\sigma))
  \end{equation*}
  so that $\mu^s_m(w)(\sigma,\widehat 1) =
  \mu^s_{|\sigma|}(w(\sigma))(\widehat 0, \widehat 1)$.  More
  generally, suppose that $\sigma < \tau$ for some $\tau \in
  L^s_m(w)$. Assume that the partition $\tau$ has blocks $\tau_j$.
  Let $\sigma_j$ be the set of those blocks of $\sigma$
  that intersect the block $\tau_j$ of $\tau$. Let $w(\sigma_j)$ be the
  restriction of $w(\sigma)$ to $\sigma_j$. Then the interval
  \begin{equation*}
    L^s_m(w) \supset [\sigma,\tau] =  \prod_j L^s_{|\sigma_j|}(w(\sigma_j))
  \end{equation*}
  and therefore the value of the M\"obius\ function on the pair
  $(\sigma,\tau)$
  \begin{equation*}
    \mu^s_m(w)(\sigma,\tau) = \prod_j
    \mu^s_{|\sigma_j|}(w(\sigma_j))(\widehat 0, \widehat 1)
  \end{equation*}
  by the product theorem for M\"obius\ functions \cite[Proposition
  3.8.2]{stanley97}. We conclude that the complete M\"obius\ functions on
  all the lattices $L^s_m(w)$, are actually determined by the values
  $\mu^s_m(w)(\widehat 0, \widehat 1)$ of these M\"obius\ functions on just
  $(\widehat 0, \widehat 1)$. See Equation \eqref{eq:umsw01} for more
  information about these Euler characteristic s.

  For the following it is convenient to name the elements of the
  domain $[m]$ of $w$ so that the element $m$ carries minimal weight.
  Assume that $a_m = (1 \cdots m-1)(m)$ is an element of $L^s_m(w)$,
  ie that $w(1)+\cdots+w(m-1)>s$. We shall determine the set of
  lattice elements $x$ with $x \wedge a_m = \widehat 0$. There is only
  one solution to this equation with $x \leq a_m$ and that is $x =
  \widehat 0$. As the other solutions satisfy $x \nleq a_m$, they must
  have a block that contains $m$ and at least one other element. It
  follows that the solutions $x \neq \widehat 0$ are all elements of
  the form
  \begin{equation*}
    x = (x_1 \cdots x_tm)(\cdot) \cdots (\cdot) \text{\ with\ }
    \begin{cases}
      w(x_1) > s-w(m) & t=1 \\
      s \geq w(x_1)+ \cdots + w(x_t) > s-w(m) & t>1
    \end{cases}
  \end{equation*}
  where all blocks but the unique block containing $m$ are singletons.
  There are $t+1$ elements in the block containing $m$ where $t$ is
  some number in the range $1 \leq t \leq s$. (All the solutions $x
  \neq \widehat 0$ are atoms in the lattice $L^s_m(w)$.) Since we are
  in a lattice, the M\"obius\ function satisfies the equation
  \cite[Corollary 3.9.3]{stanley97}
  \begin{equation*}
    \mu^s_m(w)(\widehat 0, \widehat 1) =
     -\sum_{\substack{ x \wedge a_m = \widehat 0 \\ x \neq \widehat
         0}} \mu^s_m(w)(x,\widehat 1)
  \end{equation*}
 which translates to

  \begin{multline}\label{eq:mblattice}
    \mu^s_m(w)(\widehat 0, \widehat 1)  =
    -\sum_{\substack{ x \wedge a_m = \widehat 0 \\ x \neq \widehat 0}}
    \mu^s_{|x|}(w(x))(\widehat 0, \widehat 1) = \\
    - \sum_{\substack{1 \leq x_1 \leq m-1 \\ w(x_1)>s-w(m)}}
    \mu^s_{m-1}(w(x_1m)w(\cdot)\cdots w(\cdot))(\widehat 0, \widehat 1)
    - \sum_{1 < t \leq s}
   \sum_{\substack{1 \leq x_1,\ldots,x_t \leq m-1 \\
    s \geq w(x_1)+\cdots+w(x_t) > s-w(m)}}
   \mu^s_{m-t}(w(x_1\cdots
   x_tm))w(\cdot)\cdots w(\cdot))(\widehat 0, \widehat 1)
  \end{multline}
  This describes a recursive procedure for computing all values of the
  M\"obius\ function on the weight lattices $L^s_m(w)$.

  As an illustration we compute $\mu^2_6(1^6)(\widehat 0, \widehat
  1)$. Using \eqref{eq:mblattice} twice gives
  \begin{equation*}
   \mu^2_6(1^6)(\widehat 0, \widehat 1) =
   -10 \mu^2_4(3111)(\widehat 0, \widehat 1) =
   10( \mu^2_3(411)(\widehat 0, \widehat 1) + \mu^2_2(33)(\widehat 0,
   \widehat 1) )
  \end{equation*}
  The lattices $L^2_4(411)$ and $L^2_2(33)$
   have $4$ and $2$ elements, respectively, and they look like
   \begin{center}
     \begin{tikzpicture}
  [vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  4pt, inner sep =0pt,draw}, vertex/.default=black]

  \node at (-4.5,1) {$L^2_3(411) \colon$};
  \draw [thick] (0,0)  -- (-1,1) -- (0,2) -- (1,1)  -- (0,0) -- cycle;
  \node at (0,0) [vertex, label = below:{$\mu(\widehat 0, \cdot)=1$}] {};
  \node at (-1,1) [vertex, label = left:{$\mu(\widehat 0, \cdot)=-1$}] {};
  \node at (0,2) [vertex, label = above:{$\mu(\widehat 0, \cdot)=1$}] {};
  \node at (1,1) [vertex,label = right:{$\mu(\widehat 0, \cdot)=1$}] {};

  \node at (4.5,1) {$L^2_2(33) \colon$};
  \draw [thick] (6,0) node[vertex,label = below:{$\mu(\widehat 0,
    \cdot)=1$}] {} --
  (6,2) node[vertex,label = above:{$\mu(\widehat 0, \cdot)=-1$}] {};
\end{tikzpicture}
\end{center}
so that $\mu^2_3(411)(\widehat 0, \widehat 1) = 1$ and
$\mu^2_2(33)(\widehat 0, \widehat 1) = -1$. Therefore
$\mu^2_6(1^6)(\widehat 0, \widehat 1)=0$.

\end{rmk}

We remind the reader of the well-known fact that $\mu_m^s(w)(\widehat
0, \widehat 1)$ is the reduced Euler characteristic\ of the open interval
$L^s_m(w)(\widehat 0, \widehat 1)$ between $\widehat 0$ and $\widehat
1$  in the lattice $L^s_m(w)$.

\begin{prop}\cite[\S 6]{rota64} \cite[Proposition 3.8.5]{stanley97}
  Let $x<y$ be two elements in a finite poset. The value of the M\"obius\
  function on the pair $(x,y)$ is the reduced Euler characteristic\ of the open
  interval $(x,y)$.
\end{prop}
\begin{proof}
  Write $\mu$ be the M\"obius\ function of $P$ and $\mathrm{E}$ for Euler characteristic
  . The closed interval from $x$ to $y$ has Euler characteristic\ $1$ since it has a
  smallest element. Thus
  \begin{multline*}
    1 = \mathrm{E}([x,y]) = \sum_{a,b \in [x,y]} \mu(a,b) =
     \sum_{a,b \in (x,y)} \mu(a,b) + \sum_{a \in [x,y]} \mu(a,y) +
      \sum_{b \in [x,y]} \mu(x,b) - \mu(x,y) \\
      = \mathrm{E}((x,y)) + 0 + 0 -\mu(x,y) = \mathrm{E}((x,y))-\mu(x,y)
  \end{multline*}
  or $\mu(x,y) = \widetilde{\mathrm{E}}((x,y))$.
\end{proof}

For $1 \leq s \leq m+1$ let $B(m,s)$ be the graded poset of
nonempty subsets of $[m]$ of
cardinality less than $s$.

\begin{lemma}\label{lemma:Bms}
  The reduced Euler characteristic\ of $B(m,s)$ is
  \begin{equation*}
    \widetilde E(B(m,s)) = (-1)^s \binom {m-1}{s-1},  \qquad
    1 \leq s\leq m+1
  \end{equation*}
\end{lemma}
\begin{proof}
   It is rather easy to get the recurrence relation
   \begin{align*}
    &E(B(m,2)) = m \\
    &E(B(m,s)) =
   E(B(m,s-1)) + \binom m{s-1} \sum_{j=1}^{s-1} (-1)^{s-1-j}
   \binom{s-1}j, \qquad 2<s<2+m
  \end{align*}
  Since the sum of binomial coefficients has value $(-1)^s$, we
  get the recurrence relation
  \begin{align*}
     &\widetilde E(B(m,2)) = m-1 \\
    &\widetilde E(B(m,s)) =
   \widetilde E(B(m,s-1)) + (-1)^s\binom m{s-1},  \qquad 2<s<2+m
  \end{align*}
  for the reduced Euler characteristic . The claim of the lemma follows immediately.
\end{proof}

\begin{exmp}[Reduced Euler characteristic s of the $s$-chromatic lattice intervals
  $L^s_m(w)(\widehat 0, \widehat 1)$]
  The reduced Euler characteristic s $\mu_m^s(1^m)(\widehat 0, \widehat 1) =
  \widetilde E(L^s_m(1^m)(\widehat 0, \widehat 1))$, $m \geq s+2$, for
  $s=1,2,\ldots,8$ are
\begin{align*}
&2, -6, 24, -120, 720, -5040, 40320, -362880, 3628800, -39916800,
479001600, -6227020800, 87178291200, \ldots \\
& 3, -6, 0, 90, -630, 2520, 0, -113400, 1247400, -7484400, 0,
681080400, -10216206000, 81729648000, \ldots \\
& 4, -10, 20, -70, 560, -4200, 25200, -138600, 924000, -8408400,
84084000, -798798000, 7399392000, \ldots \\
& 5, -15, 35, -70, 0, 2100, -23100, 173250, -1051050, 5255250,
-15765750, -105105000, 2858856000,\ldots \\
& 6, -21, 56, -126, 252, -924, 11088, -126126, 1093092,
-7693686, 46414368, -254438184, 1492322832, \ldots \\
& 7, -28, 84, -210, 462, -924, 0, 42042, -630630,
6390384, -51459408, 351639288, -2118412296,11406835440\ldots \\
&   8, -36, 120, -330, 792, -1716, 3432, -12870,
205920, -3150576, 35706528,-322583976,2460949920 \ldots \\
&  9, -45, 165, -495, 1287, -3003, 6435, -12870, 0,
787644, -14965236,191222460, -1920538620 \ldots
\end{align*}
The first sequence, $\mu_m^1(1^m)(\widehat 0,
\widehat 1)$, $m \geq 2$, is the sequence $(-1)^{m-1}(m-1)!$ of
reduced Euler characteristic s of the lattice of partitions of $[m]$ \cite[Example
3.10.4]{stanley97}.  The second sequence, $\mu_m^2(1^m)(\widehat 0,
\widehat 1)$, $m \geq 3$, seems to coincide with first terms of the
sequence \href{http://oeis.org/A009014}{A009014} from The On-Line
Encyclopedia of Integer Sequences (OES). The remaining $6$ sequences
apparently do not match any sequences of the OES.

The first $s$ terms of these sequences are signed binomial
coefficients. This is because the interval $(\widehat 0, \widehat 1)$
in $L^s(D[m])$ is isomorphic to the opposite of the poset $B(m,m-s)$
when $s+2 \leq m \leq 2s+1$. Thus the reduced Euler characteristic\
\begin{equation*}
  \mu_m^s(1^m)(\widehat 0, \widehat 1) = \widetilde E (B(m,m-s)) =
  (-1)^{m-s} \binom {m-1}s, \qquad s+2 \leq m \leq 2s+1,
\end{equation*}
according to Lemma~\ref{lemma:Bms}.

The first terms of the sequence $\mu_m^2(3^11^{m-1})(\widehat 0,
\widehat 1)$, $m \geq 3$, of reduced Euler characteristic s of the weighted lattice intervals
$L^2_m(3^11^{m-1})(\widehat 0,
\widehat 1)$,
\begin{equation*}
  1, 0, -6, 30, -90, 0, 2520, -22680, 113400, 0, -7484400, 97297200,
  -681080400, 0, 81729648000, -1389404016000, \ldots
\end{equation*}
seem to coincide up to sign with first terms of the sequence
\href{http://oeis.org/A009775}{A009775} from OES.  The sequence of
reduced Euler characteristic s $\mu_m^2(3^21^{m-2})(\widehat 0, \widehat 1)$, $m \geq
3$, of the lattice interval $L^2_m(3^21^{m-2})(\widehat 0, \widehat
1)$,
\begin{multline*}
2, -4, 6, 6, -120, 720, -2520, -2520, 136080, -1360800, 7484400,
7484400, \\ -778377600, 10897286400, -81729648000, -81729648000,
 13894040160000, \ldots
\end{multline*}
apparently does not match any sequence in the OES.
\end{exmp}

Define the {\em $s$-monochrome set\/} of a map $\mathrm{col} \colon
V(K) \to [r]=\{1,\ldots,r\}$ to be the set
\begin{equation*}
  M^s(\mathrm{col}) = \{ \sigma \in F^s(K) \mid |\mathrm{col}(\sigma)| = 1 \}
\end{equation*}
of all monochrome $s$-simplices in $K$.  The map $\mathrm{col}$ is an
$(r,s)$-coloring of $K$ if and only if
$M^s(\mathrm{col})=\emptyset$.

\begin{lemma}\label{lemma:Mscol}
  The $s$-monochrome set $M^s(\mathrm{col})$ of any map $\mathrm{col}
  \colon V(K) \to [r]$ is an element of the $s$-chromatic lattice
  $L^s(K)$.
\end{lemma}
\begin{proof}
  Let $S$ be a connected component of $M^s(\mathrm{col})$. Since $S$ is
  connected, all vertices in $S$ have the {\em same\/} color. Let
  $\sigma \in F^s(K)$ be an $s$-simplex of $K$ such that $\sigma
  \subset V(S)$. The $\sigma$ is monochrome: $\sigma \in
  M^s(\mathrm{col})$. By Lemma~\ref{lemma:closedcomp}, $S$ is closed.
\end{proof}

\begin{thm}\label{thm:LsK}
  The number of $(r,s)$-colorings of $K$ is
  \begin{equation*}
    \chrps Krs=\sum_{T \in L^s(K)} \mu(\widehat 0,T)  r^{|\pi(T)|}
  \end{equation*}
 where $\mu$ the M\"obius\ function for the $s$-chromatic lattice $L^s(K)$.

\end{thm}
\begin{proof}
  For any $B \in L^s(K)$, let $\chi(K,r,s,B)$ be the number of maps
  $\mathrm{col} \colon V(K) \to [r]$ with $M^s(\mathrm{col})=B$.  We
  want to determine $\chi(K,r,s,\emptyset) = \chrps sKr$.  For any $A
  \in L^s(K)$,
  \begin{equation*}
     r^{|\pi(A)|} =
     \sum_{A \leq B} \chi(K,r,s,B)
  \end{equation*}
  because there are $r^{|\pi_0(A)|} r^{m(K)-|V(A)|} =
  r^{|\pi(A)|}$ maps $\mathrm{col} \colon V(K) \to [r]$ with $A
  \leq M^s(\mathrm{col})$.  Equivalently,
  \begin{equation*}
    \sum_{A \leq B} \mu(A,B)  r^{|\pi(B)|} = \chi(K,r,s,A)
  \end{equation*}
  by M\"obius\ inversion \cite[Proposition 3.7.1]{stanley97}. The statement
  of the theorem is the particular case of this formula where
  $A=\widehat 0$.
\end{proof}

The  defining rules for the M\"obius\ function of the poset $L^s(K)$
\cite[3.7]{stanley97}
\begin{itemize}
\item $\mu(S,S)=1$ for all $S \in L^s(K)$
\item $\sum_{R \leq S \leq T} \mu(R,S)=0$ when $R \lneqq T$
\item $\mu(R,S)=0$ when $R \nleq S$
\end{itemize}
imply that $\mu(\widehat 0,\widehat 0)=1$ and
$\mu(\widehat 0,\{\sigma\})=-1$ for every $s$-simplex $\sigma \in
F^s(K)$.

\begin{cor}\label{cor:LsK}
  The highest degree terms of the $s$-chromatic polynomial are
  \begin{equation*}
    \chrps Krs = r^{m(K)} - f_s(K)r^{m(K)-s} + \cdots
  \end{equation*}
  Thus the  $s$-chromatic polynomial determines $f_0(K)$ and $f_s(K)$.
\end{cor}
\begin{proof}
  The $s$-chromatic polynomial is
  \begin{equation*}
    \chrps Krs = \mu(\widehat 0,\widehat 0)r^{f_0(K)} +
        \sum_{\sigma \in F^s(K)}\mu(\widehat 0,\{\sigma\})r^{f_0(K)-s}
        + \cdots
  \end{equation*}
  where $\mu(\widehat 0,\widehat 0)=1$ and  $\mu(\widehat
  0,\{\sigma\})=-1$ for all $s$-simplices $\sigma$ of $K$.
\end{proof}

\begin{example}
  Consider the $2$-dimensional complex $K$ from Example~\ref{exmp:K}.
The $2$-chromatic lattice $L^2(K)$ of $K$
\begin{equation*}
   \xy 0;/r.20pc/:
  (0,0)*@{*}="1"+(0,-5)*{\{1,2,3\}};
  (20,0)*@{*}="2"+(0,-5)*{\{4,5,6\}};
  (40,0)*@{*}="3"+(0,-5)*{\{2,3,4\}};
  (0,20)*@{*}="4"+(0,+5)*{
    \begin{array}{c}
      \{1,2,3\} \\ \{4,5,6\}
    \end{array}};
  (20,20)*@{*}="5"+(0,+5)*{
    \begin{array}{c}
      \{1,2,3\} \\ \{2,3,4\}
    \end{array}};
  (40,20)*@{*}="6"+(0,+5)*{
    \begin{array}{c}
      \{2,3,4\} \\ \{4,5,6\}
    \end{array}};
  (20,-20)*@{*}="0"+(0,-5)*{\widehat 0};
  (20,40)*@{*}="7"+(0,+5)*{\widehat 1};
   "1"; "4" **\dir{-};
   "1"; "5" **\dir{-};
   "2"; "4" **\dir{-};
   "2"; "6" **\dir{-};
   "3"; "5" **\dir{-};
   "3"; "6" **\dir{-};
   "0"; "1" **\dir{-};
   "0"; "2" **\dir{-};
   "0"; "3" **\dir{-};
   "7"; "4" **\dir{-};
   "7"; "5" **\dir{-};
   "7"; "6" **\dir{-};
   (90,-20)*{\mu(S)=+1 \qquad |\pi(S)| = 6\phantom{,4,4} };
   (90,0)*{\mu(S)=-1 \qquad |\pi(S)| = 4,4,4};
   (90,20)*{\mu(S)=+1 \qquad |\pi(S)| = 2,3,2};
   (90,40)*{\mu(S)=-1 \qquad |\pi(S)| = 1\phantom{,4,4}};
  \endxy
\end{equation*}
consists of {\em all\/} subsets of $F^2(K)$.  The $2$-chromatic
polynomial is
  \begin{equation*}
    \chrps Kr2 = r^6 - r^4 - r^4 -r^4 + r^2 + r^3 + r^2 - r =
           r^6 - 3r^4 + r^3 + 2r^2 -r

  \end{equation*}
  $K$ has $\chrps K22 = 30$ $(2,2)$-colorings and $\chrps K32 = 528$
  $(3,2)$-colorings.
\end{example}

\begin{exmp}\label{exmp:MB}
The triangulation $\mathrm{MB}$ of the M\"obius\ band with $f$-vector
$f(\mathrm{MB}) = (5,10,5)$ shown in Figure~\ref{fig:MBcol}
has the following (reduced) $2$-chromatic lattice
$L^2(\mathrm{MB})-\{\widehat 0, \widehat 1\}$
  \begin{equation*}
   \xy 0;/r.20pc/:
  (0,0)*@{*}="2"+(0,-5)*{\{1,3,5\}};
  (20,0)*@{*}="4"+(0,-5)*{\{2,3,5\}};
  (40,0)*@{*}="3"+(0,-5)*{\{1,3,4\}};
  (60,0)*@{*}="5"+(0,-5)*{\{2,4,5\}};
  (80,0)*@{*}="6"+(0,-5)*{\{1,2,4\}};

  (0,20)*@{*}="11"+(0,5)*{
    \begin{array}[c]{c}
      \{1,3,5\} \\ \{2,3,5\}
    \end{array}};
  (20,20)*@{*}="10"+(0,5)*{
    \begin{array}[c]{c}
      \{1,3,5\} \\ \{1,3,4\}
    \end{array}};
  (40,20)*@{*}="8"+(0,5)*{
    \begin{array}[c]{c}
      \{2,4,5\} \\ \{2,3,5\}
    \end{array}};
  (60,20)*@{*}="9"+(0,5)*{
    \begin{array}[c]{c}
      \{1,2,4\} \\ \{1,3,4\}
    \end{array}};
  (80,20)*@{*}="7"+(0,5)*{
    \begin{array}[c]{c}
      \{2,4,5\} \\ \{1,2,4\}
    \end{array}};

   "5"; "7" **\dir{-};
   "6"; "7" **\dir{-};

   "4"; "8" **\dir{-};
   "5"; "8" **\dir{-};

   "3"; "9" **\dir{-};
   "6"; "9" **\dir{-};

   "2"; "10" **\dir{-};
   "3"; "10" **\dir{-};

   "2"; "11" **\dir{-};
   "4"; "11" **\dir{-};
   (130,0)*{\mu=-1 \qquad |\pi(S)| = 3,3,3,3,3};
   (130,20)*{\mu=+1 \qquad |\pi(S)| = 2,2,2,2,2};
  \endxy
  \end{equation*}
  and $2$-chromatic polynomial
  \begin{equation*}
    \chrps {\mathrm{MB}}r2 = r^5 - 5r^3 + 5r^2 -r

  \end{equation*}
  The lattice $L^2(\mathrm{MB})$ is graded but it is still not
  semi-modular \cite[Proposition 3.3.2]{stanley97}: The meet and join
  of $a=\{\{2,3,5\}\}$ and $b=\{\{1,3,4\}\}$ are $a \wedge b =
  \widehat 0$ and $a \vee b = \widehat 1$. Thus $a$ and $b$ cover $a
  \wedge b$ but $a \vee b$ covers neither $a$ nor $b$.
\end{exmp}

\begin{figure}[t]
  \centering
\begin{tikzpicture}
  [vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  8pt, inner sep =0pt,draw}, vertex/.default=black, scale=1.25]

\begin{scope}[xshift=-4.5cm,yshift=1.75cm]

\draw (0,0) circle (2cm);

\coordinate [vertex=blue, label=right:$1$] (A1) at (0:2cm);
\coordinate [vertex=red, label=above right:$2$] (A2) at (60:2cm);
\coordinate [vertex=yellow, label=above left:$3$] (A3) at (120:2cm);

\coordinate [vertex=blue, label=left:$1$] (B1) at (-180:2cm);
\coordinate [vertex=red, label=below left:$2$] (B2) at (-120:2cm);
\coordinate [vertex=yellow, label=below right:$3$] (B3) at (-60:2cm);

\begin{scope}[rotate=-60]
\coordinate [vertex=blue, label=0:$4$] (C4) at (240:1cm);
\coordinate [vertex=red, label=266:$5$] (C5) at (120:1cm);
\coordinate [vertex=yellow, label=100:$6$] (C6) at (0:1cm);
\end{scope}

\draw (A1) -- (C5) -- (C6) -- (A1) -- cycle;
\draw (A2) -- (C5) -- (A1);
\draw (A2) -- (C5) -- (A3);
\draw (A3) -- (C4) -- (C5) -- (A3);
\draw (A3) -- (C4) -- (B1);
\draw (B2) -- (C4) -- (C6) -- (B2);
\draw (B2) -- (C6) -- (B3);
\end{scope}

\begin{scope}[scale=1.2]
\coordinate [vertex=red, label=below:$1$] (A1) at (-.75,0);
\coordinate [vertex=blue, label=below:$4$] (A2) at (.75,0);
\coordinate [vertex=yellow, label=below:$7$] (A3) at (2.25,0);
\coordinate [vertex=red, label=below:$3$] (A4) at (3.75,0);

\coordinate [vertex=red, label=left:$3$] (B2) at (0,1);
\coordinate [vertex=red, label=left:$2$] (B3) at (0,2);
\coordinate [vertex=red, label=above:$1$] (B4) at (-.75,3);

\coordinate [vertex=blue, label=above:$4$] (C2) at (.75,3);
\coordinate [vertex=yellow, label=above:$7$] (C3) at (2.25,3);
\coordinate [vertex=red, label=above:$3$] (C4) at (3.75,3);

\coordinate [vertex=red, label=right:$2$] (D2) at (3,1);
\coordinate [vertex=red, label=right:$1$] (D3) at (3,2);

\coordinate [vertex=blue, label=above:$5$] (E1) at (1.5,2);
\coordinate [vertex=blue, label=below:$6$] (E2) at (1.5,1);

\draw (A1) -- (A2) -- (A3) -- (A4) -- (D2) -- (D3) -- (C4) -- (C3) --
(C2) -- (B4) -- (B3) -- (B2) -- (A1);

\draw (C2) -- (C3) -- (E1) -- (C2);
\draw (A2) -- (A3) -- (E2) -- (A2);
\draw (D3) -- (E1) -- (E2) -- (D3);
\draw (D2) -- (E2) -- (A3) -- (D2);
\draw (C3) -- (D3);
\draw (B3) -- (C2);
\draw (B3) -- (E1);
\draw (B2) -- (E1);
\draw (B2) -- (E2);
\draw (B2) -- (A2);
\draw [dotted,<->,gray] (A4) .. controls (4.25,0.5) and (4.25,2.5) .. (C4);
\draw [dotted,<->,gray] (A1) .. controls (-1.25,0.5) and (-1.25,2.5)..  (B4);
\end{scope}
\end{tikzpicture}
\caption{$(3,2)$-colorings of $\mathrm{P2}$ and $\mathrm{MT}$}
  \label{fig:P2col}
\end{figure}

\begin{exmp}\label{exmp:MT}

  Let $\mathrm{MT}$ be M\"obius 's minimal triangulation of the torus with
  $f$-vector $f(\mathrm{MT})=(7, 21, 14 )$ and $\mathrm{P}2$ the
  triangulation of the projective plane with $f$-vector
  $f(\mathrm{P}2) = (1,6,15,10)$ shown in Figure~\ref{fig:P2col}
  (decorated with $(3,2)$-colorings).  The chromatic polynomial s of these two
  simplicial complexes are
  \begin{alignat*}{3}
    &\chrps {\mathrm{MT}}r1 = [r]_7, & &\qquad &
    &\chrps {\mathrm{MT}}r2 = r^7 - 14r^5 + 21r^4 + 7r^3 - 21r^2 + 6r

    \\
    &\chrps {\mathrm{P}2}r1 = [r]_6, & &\qquad &
    & \chrps {\mathrm{P}2}r2 = r^6 - 10r^4 + 15r^3 - 6r^2
  \end{alignat*}
  In both cases, the $1$-skeleton is the complete graph on the vertex
  set. The chromatic numbers are $\chs 1{\mathrm{MT}} = 7$, $\chs
  1{\mathrm{P}2} = 6$, and $\chs 2{\mathrm{MT}} = 3 = \chs
  2{\mathrm{P}2}$.

\end{exmp}

  The chromatic polynomial s of  simple graphs (the $1$-chromatic polynomial s of simplicial complexes)
  are known to have these properties:
  \begin{itemize}
  \item The coefficients are sign-alternating \cite[\S7,
    Corollary]{rota64}
  \item  The coefficients are $\log$-concave
  (Definition~\ref{defn:logconcave}) in absolute value \cite{huh2012}
\item There are no negative roots and no roots between $0$ and $1$
  \cite{woodall77}
  \end{itemize}
  In contrast, the coefficients of the $2$-chromatic polynomial\
  \begin{equation*}
     \chrps {\mathrm{MT}}r2 =  r^7 - 14r^5 + 21r^4 + 7r^3 - 21r^2 + 6r =
     [r]_3(r+1)(r^3 + 2r^2 - 9r + 3)
  \end{equation*}
  are not sign-alternating, not $\log$-concave in absolute value, and
  the polynomial has a negative root and a root between $0$ and $1$.

\subsection{The $s$-chromatic polynomial\ in falling factorial form}
\label{sec:FF}

Theorem~\ref{lemma:StK} provides an interpretation of the coefficients
of the falling factorial $[r]_i$ in the $s$-chromatic polynomial of
the simplicial complex $K$.

\begin{defn}\label{defn:SKrs}
$S(K,r,s)$ is the number of partitions of $V(K)$ into $r$
$s$-independent blocks.
\end{defn}

We think of $S(K,r,s)$ as an $s$-Stirling number of the second kind
for the simplicial complex $K$. If $s > \dim(K)$, then there are no
$s$-simplices in $K$ and all partitions of $V(K)$ are $s$-independent,
so that $S(K,r,s)$ is the Stirling number of the second kind
$S(m(K),r)$ \cite[p $33$]{stanley97}.  We now explain the general
relation between these simplicial Stirling numbers $S(K,r,s)$ and the
usual Stirling numbers of the second kind.

Define the {\em $s$-monochrome set\/} of a partition $P$ of $V(K)$ to
be the set
\begin{equation*}
  M^s(P) = \{ \sigma \in F^s(K) \mid \text{$\sigma$ is contained in a
    block of $P$} \}
\end{equation*}
of all $s$-simplices entirely contained in one of the blocks of $P$.
The set $M^s(P)$ is an element of the $s$-chromatic lattice $L^s(K)$
by Lemma~\ref{lemma:Mscol}.

\begin{thm}\label{thm:SKrs}
  The number of partitions of $V(K)$ into $r$ $s$-independent blocks
  is
  \begin{equation*}
    S(K,r,s) = \sum_{T \in L^s(K)} \mu(\widehat 0,T) S(|\pi(T)|,r)
  \end{equation*}
 where $\mu$ the M\"obius\ function for the $s$-chromatic lattice $L^s(K)$.
\end{thm}
\begin{proof}
  For any $B \in L^s(K)$, let $S(K,r,s,B)$ be the number of partitions
  $P$ of $V(K)$ into $r$ blocks with monochrome set $M^s(P)=B$.
  We want to determine $S(K,r,s,\emptyset) = S(K,r,s)$.
  For
  any $A \in L^s(K)$,
  \begin{equation*}
    S(|\pi(A)|,r) = \sum_{A \leq B} S(K,r,s,B)
  \end{equation*}
  because there are $S(|\pi(A)|,r)$ partitions $P$ of $V(K)$ into $r$
  blocks with $A \leq M^s(P)$. Equivalently,
  \begin{equation*}
      \sum_{A \leq B} \mu(A,B) S(|\pi(B)|,r) = S(K,r,s,A)
  \end{equation*}
  by M\"obius\ inversion \cite[Proposition 3.7.1]{stanley97}. The statement
  of the theorem is the particular case of this formula where
  $A=\widehat 0$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{lemma:StK}]
  We simply follow the proof of the similar statement for chromatic polynomial s for
  graphs \cite[Theorem 15]{read68}.  When $r \geq i$ we can get an
  $(r,s)$-coloring out of one of the $S(K,i,s)$ partitions of $V(K)$
  into $i$ $s$-independent blocks by choosing $i$ out of the $r$
  colors and assigning them to the $i$ blocks. There are $\binom ri$
  ways of choosing the $i$ out of $r$ colors and $i!$ ways of coloring
  $i$ blocks in $i$ colors.  The number of $(r,s)$-colorings of $K$ in
  exactly $i$ colors is thus
\begin{equation*}
  S(K,i,s) \binom ri i!  = S(K,i,s) [r]_i
\end{equation*}
so that
\begin{equation*}
  \chrps Krs = \sum_{i=1}^{m(K)}  S(K,i,s) [r]_i
\end{equation*}
is the total number of $(r,s)$-colorings of $K$.
\end{proof}

\begin{cor}\label{cor:chiLsK}
  The reduced Euler characteristic\ of the open interval $(\widehat 0, \widehat 1)$ in
  $s$-chromatic lattice $L^s(K)$ is
  \begin{equation*}
    \mu(L^s(K))(\widehat 0, \widehat 1) =
    \sum_{i=\chs sK}^{m(K)} (-1)^{i-1}(i-1)!S(K,i,s)
  \end{equation*}
\end{cor}
\begin{proof}
  Equate the terms of degree $1$ of the two expressions
\begin{equation}\label{eq:twochisKr}
    \sum_{T \in L^s(K)} \mu(\widehat 0, T) r^{|\pi(T)|}
     = \sum_{i=\chs sK}^{m(K)} S(K,i,s)[r]_i
  \end{equation}
  from Theorem~\ref{thm:LsK} and Theorem~\ref{lemma:StK} for the
  $s$-chromatic polynomial\ of $K$.
\end{proof}

We observe that
\begin{equation*}
   \sum_i S(K,i,s)[r]_i
             = \sum_i \sum_T \mu(\widehat 0,T) S(|\pi(T)|,i)[r]_i
             = \sum_T \mu(\widehat 0,T) \sum_i  S(|\pi(T)|,i)[r]_i
             =  \sum_T \mu(\widehat 0,T) r^{|\pi(T)|}
\end{equation*}
so that Theorem~\ref{thm:SKrs} implies Theorem~\ref{lemma:StK}.

The $s$-chromatic number of $K$ is immediately visible
with the $s$-chromatic polynomial\ in factorial form because
\begin{equation*}
  \chs sK = \min \{ i \mid S(K,i,s) \neq 0 \}
\end{equation*}
is the lowest degree of the nonzero terms. The positive integer
sequence
\begin{equation*}
  \chrps K{\chs sK}s ,\ldots,\chrps K{m(K)}s = 1
\end{equation*}
has no internal zeros. (If there is a partition of $V(K)$ into $r$
blocks not containing any $s$-simplex of $K$ and $r<m(K)$, then
split one of the blocks with more than one vertex into two sub-blocks
to get a partition of $V(K)$ into $r+1$ blocks containing no
$s$-simplices of $K$.)

The simplicial Stirling numbers satisfy the recurrence relations
\begin{equation*}
 S(K,r,s) =  \sum_{\substack{\emptyset \subsetneq U \subseteq V(K)-\{
     v_0\} \\
     \text{$V(K)-U$ $s$-independent}}} S(K \cap D[U],r-1,s), \qquad
S(K,1,s) =
\begin{cases}
  1 & s> \dim(K) \geq 0  \\ 0 & \text{otherwise}
\end{cases}
\end{equation*}
To see this, fix a vertex $v_0$ of $K$.
Let $P$ be  partition of $V(K)$ into $r$ $s$-independent
subsets. Let $U_0$ be the block containing $v_0$. The other blocks in
$P$ form a partition $P_0$
of $K \cap D[V(K)-U_0]$ into $r-1$ $s$-independent
subsets. The map $P \leftrightarrow (P_0,U_0)$ is a bijection.

The familiar recurrence relation $S(m,r) = S(m-1,r-1)+rS(m-1,r)$ for
Stirling numbers of the second kind does not readily apply to
simplicial Stirling numbers. The closest analogue may be
\begin{equation*}
  S(K,r,s) = S(K \cap D[V(K)-\{v_0\}],r-1,s) +
  \sum_{P \in  \mathcal{S}(K \cap D[V(K)-\{v_0\}],r,s)}
  |\{ B \in P \mid B \cup \text{$\{v_0\}$ is $s$-independent in $K$} \}|
\end{equation*}
where $v_0$ is a vertex of $K$ and $ \mathcal{S}(K \cap
D[V(K)-\{v_0\},r,s)$ is the set of partitions $P$ of the vertex set of $K
\cap D[V(K)-\{v_0\}$ into $r$ $s$-independent subsets.

\begin{prop}\label{prop:SKisSNis}
Let $K$ be a subcomplex of $L$  and assume that $V(K)=V(L)$.
\begin{enumerate}
\item $S(K,r,s) \geq S(L,r,s)$ for all $r$.\label{prop:SKisSNis1}
\item If $S(K,r,s) = S(L,r,s)$ for some $r$ with
  $\frac{1}{s}(|V|-1) \leq r \leq |V|-s$, then $K^s = L^s$.
  \label{prop:SKisSNis2}
\end{enumerate}
\end{prop}
\begin{proof}
  \noindent \eqref{prop:SKisSNis1} Let $V$ be the vertex set of $K$ and
  $L$. Write $\mathcal{S}(K,r,s)$ and $\mathcal{S}(L,r,s)$ for the set
  of partitions of $V$ into $r$ blocks containing no $s$-simplex of
  $K$ or $L$, respectively. Then $\mathcal{S}(L,r,s) \subseteq
  \mathcal{S}(K,r,s)$ for all $r$ and $s$. Thus $S(L,r,s) \leq
  S(K,r,s)$.

  \noindent \eqref{prop:SKisSNis2} Suppose that $\sigma \in
  F^s(L)-F^s(K)$ is an $s$-simplex of $L$ that is not an $s$-simplex
  of $K$. Any partition of the form
  \begin{equation*}
    \{\sigma\} \cup \tau, \qquad \tau \in
    \mathcal{S}(D[V-\sigma],r-1,s),
  \end{equation*}
  in $\mathcal{S}(K,r,s)-\mathcal{S}(L,r,s)$. The set
  $\mathcal{S}(D[V-\sigma],r-1,s)$ is nonempty when
  \begin{equation*}
    \chs s{D[V-\sigma]} =
    \left\lceil \frac{|V|-s-1}{s} \right\rceil \leq r-1 \leq |V|-s-1
  \end{equation*}
  and thus $S(K,r,s)$ is strictly greater than $S(L,r,s)$ when
  $\frac{|V|-1}{s} \leq r \leq |V|-s$.
\end{proof}

\begin{rmk}[$S(K,r,s)$ for the complete simplex {$K=D[m]$}]\label{rmk:Smrs}

  For any finite set $M$, let $S(M,r,s)$ stand for $S(D[M],r,s)$
  (Definition~\ref{defn:SKrs}), the number of partitions of the set
  $M$ into $r$ blocks containing at most $s$ elements. Let us even
  write $S(m,r,s)$ in case $M=[m]$, $m \geq 1$, $r,s \geq 0$.

  Clearly, $S(m,r,s)$ is nonzero only when $m/s \leq r \leq m$. Also,
  $S(m,r,s) = S(m,r)$ when $r$ is among the $s$ numbers
  $m-s+1,\ldots,m$.

The recurrence relation
\begin{equation*}
  S(m,r,s) = \sum_{j=m-s}^{m-1} \binom {m-1}j S(j,r-1,s)

\end{equation*}
can be used to compute these numbers.  Table~\ref{tab:chrtableDm}
shows $S(m,r,s)$ for small $m$; the number $S(m,r,s)$ is in row $s$
and column $r$ in the chromatic table (Definition~\ref{defn:chitable})
for $D[m]$.  All the red numbers are usual Stirling numbers of the
second kind.

\begin{table}[t]
  \centering
  \begin{tabular}[t]{*{2}{>{$}c<{$}}}
    \begin{pmatrix}
      \textcolor{blue}{0} & \textcolor{red}{1}
    \end{pmatrix} &
    \begin{pmatrix}
      0 & \textcolor{blue}{0} & \textcolor{red}{1} \\
      \textcolor{blue}{0} & \textcolor{red}{3}
      & \textcolor{red}{1}
    \end{pmatrix} \\
    \begin{pmatrix}
      0 & 0 & \textcolor{blue}{0} & \textcolor{red}{1} \\
      0 & \textcolor{blue}{3} & \textcolor{red}{6} & \textcolor{red}{1} \\
      \textcolor{blue}{0} & \textcolor{red}{7}
      & \textcolor{red}{6} & \textcolor{red}{1}
    \end{pmatrix} &
    \begin{pmatrix}
       0 & 0 & 0  & \textcolor{blue}{0} & \textcolor{red}{1} \\
       0 & 0 & \textcolor{blue}{15}
       & \textcolor{red}{10} &  \textcolor{red}{1} \\
       0 & \textcolor{blue}{10} & \textcolor{red}{25}
      & \textcolor{red}{10} &  \textcolor{red}{1} \\
       \textcolor{blue}{0}  & \textcolor{red}{15} & \textcolor{red}{25}
   & \textcolor{red}{10} & \textcolor{red}{1}

    \end{pmatrix} \\
    \begin{pmatrix}
     0 & 0 & 0 & 0 & \textcolor{blue}{0} & \textcolor{red}{1} \\
     0 & 0 & 15 & \textcolor{blue}{45}
     & \textcolor{red}{15} & \textcolor{red}{1} \\
     0  & 10 & \textcolor{blue}{75} & \textcolor{red}{65}
     & \textcolor{red}{15} &  \textcolor{red}{1} \\
     0 & \textcolor{blue}{25} & \textcolor{red}{90} & \textcolor{red}{65}
     & \textcolor{red}{15} &  \textcolor{red}{1} \\
     \textcolor{blue}{0} & \textcolor{red}{31}
     & \textcolor{red}{90} & \textcolor{red}{65}
     & \textcolor{red}{15} &  \textcolor{red}{1}

    \end{pmatrix} &
    \begin{pmatrix}
      0 &  0 &   0 &  0 &  0 &  \textcolor{blue}{0} &  \textcolor{red}{1} \\
      0 &  0 &  0 & 105 & \textcolor{blue}{105}
      &  \textcolor{red}{21} &   \textcolor{red}{1} \\
      0 &  0 & 175
      & \textcolor{blue}{315} & \textcolor{red}{140} &  \textcolor{red}{21}
      &  \textcolor{red}{1} \\
      0 & 35 & \textcolor{blue}{280} & \textcolor{red}{350}
      & \textcolor{red}{140}
      &  \textcolor{red}{21} &   \textcolor{red}{1} \\
      0 & \textcolor{blue}{56} & \textcolor{red}{301}
      & \textcolor{red}{350} & \textcolor{red}{140}
      &  \textcolor{red}{21} &  \textcolor{red}{1} \\
      \textcolor{blue}{0} & \textcolor{red}{63}
      & \textcolor{red}{301} & \textcolor{red}{350}
      & \textcolor{red}{140} &  \textcolor{red}{21} &  \textcolor{red}{1}

    \end{pmatrix}
  \end{tabular}
  \caption{Chromatic tables for complete simplices $D[m]$ for $m=2,\ldots,7$ }
  \label{tab:chrtableDm}
\end{table}

According to Theorem~\ref{lemma:StK}, the numbers
$S(m,r,s)$ determine the $s$-chromatic polynomial\ in falling factorial form
of the complete simplex on $m$ vertices
  \begin{equation*}
    \chrps {D[m]}rs =
    \sum_{i=\lceil m/s \rceil}^m S(m,i,s) [r]_i
  \end{equation*}
and, according to Corollary~\ref{cor:chiLsK}, they also determine the
reduced Euler characteristic\

    \begin{equation*}
       \mu_m^s(1^m)(\widehat 0, \widehat 1) =
         \sum_{i=\lceil m/s \rceil}^m (-1)^{i-1}(i-1)! S(m,i,s)

    \end{equation*}
   of the $s$-chromatic lattice $L^s(D[m])$.

   More generally, if $w \colon M \to \mathbf{N}$ is a function on $M$ with
   natural numbers as values, let $S(M,w,r,s)$ be the number of
   partitions of $M$ into admissible blocks, where we declare a block
   admissible if it is a singleton or it has weight at most $s$. (Then
   $S(m,r,s) = S([m],1^m,r,s)$ occur when $M=[m]$ and $w=1^m$ places
   weight $1$ on all elements.)  Any such partition is a partition of
   $M$ into blocks of weight at most $s$, and therefore $S(M,w,r,s)
   \leq S(\# M,r,s)$.  In particular, $S(M,w,r,s)$ is nonzero only
   when $\# M/s \leq r \leq \# M$. The recurrence relation
    \begin{equation*}
      S(M,w,r,s) =
      \sum_{\substack{\emptyset \neq J \subset M - \{\max(M)\} \\
    \text{$M-J$ admissible}}} S(J,w\vert J,r-1,s)
    \end{equation*}
    provides a means  to compute these numbers.

    The weighted version of Equation~\eqref{eq:twochisKr} for
    $K=D[m]$,
    \begin{equation*}
       \sum_{\sigma \in L^s_m(w)}
      \mu_m^s(w)(\widehat 0, \sigma)r^{|\sigma|} =
      \sum_{i= \lceil m/s \rceil}^m S([m],w,i,s) [r]_i
    \end{equation*}
    implies, by equating coefficients of first degree terms, the
    expression
    \begin{equation}\label{eq:umsw01}
      \mu_m^s(w)(\widehat 0, \widehat 1) =
       \sum_{i= \lceil m/s \rceil}^m (-1)^{i-1}(i-1)!S([m],w,i,s)
    \end{equation}
    for the Euler characteristic\ of the weighted lattice $L^s_m(w)$ from
    Remark~\ref{rmk:wlattice}.
  \end{rmk}

  Because any simplicial complex $K$ is a subcomplex of the complete
  simplex $D[m(K)]$ on its vertex set, we have
  \begin{equation}\label{eq:bounds}
    S(m(K),r) \geq S(K,r,s) \geq S(m(K),r,s),  \qquad 1 \leq r \leq m(K)

  \end{equation}
  Moreover, these inequalities are equalities for the $s$ highest
  values $m(K)-s+1, \ldots, m(K)$ of $r$. Thus the $s$ terms of
  highest falling factorial degree in the $s$-chromatic polynomial\ of $K$
\begin{equation*}
  \chrps Krs = \sum_{i=0}^{m(K)-s} S(K,i,s)[r]_i +
                \sum_{i=m(K)-s+1}^{m(K)} S(m(K),i)[r]_i
\end{equation*}
are given by the $s$ Stirling numbers $S(m(K),m(K)-s+1), \ldots,
S(m(K),m(K))$ of the second kind. These coefficients depend only
on the size of the vertex set of $K$. We shall next show that the
coefficient number $s+1$ counted from above, $S(K,m(K)-s,s)$,
informs about the number $f_s(K)$ of $s$-simplices in $K$.

\begin{prop}\label{prop:S(K,m-s,s)}
  $S(K,m(K)-s,s) = S(m(K),m(K)-s)-f_s(K)$. If $S(K,m(K)-s,s) =
  S(m(K),m(K)-s,s)$ then $K^s = D[m(K)]^s$.
\end{prop}
\begin{proof}
  The only partitions of the $S(m,m-s)$ partitions of $V(K)$ into
  $m-s$ blocks that are not $s$-independent are those consisting of
  one $s$-simplex of $K$ together with singleton blocks.  If
  $S(K,m(K)-s,s) = S(D[m(K)],m(K)-s,s)$ then $f_s(K) = f_s(D[m(K)])$
  so $K^s=D[m(K)]^s$. (This is a special case of
  Proposition~\ref{prop:SKisSNis}.\eqref{prop:SKisSNis2}.)
\end{proof}

\begin{defn}\label{defn:chitable}
  The {\em chromatic table}, $\chi(K)$, of $K$ is the $(\dim(K) \times
  m(K))$-table with $S(K,r,s)$ in row $s$ and column $r$.
\end{defn}

This means that row $s$ in the chromatic table lists the coefficients
of the $s$-chromatic polynomial . The chromatic table of a $3$-dimensional simplicial
complex $K$, for instance, looks like this
\begin{center}
  \begin{tabular}[t]
  {>{$}c<{$}|*{8}{ >{$}c<{$} }}
  {} & r=1 & r=2 & \cdots  &r=m-3 & r=m-2 & r=m-1 & r=m  \\ \hline
  S(K,\cdot,1) & S(K,1,1)  & S(K,2,1) & \cdots & S(K,m-3,1)
  & S(K,m-2,1) &
  \textcolor{blue}{S(m,m-1)-f_1} & \textcolor{red}{S(m,m)=1} \\
   S(K,\cdot,2) & S(K,1,2)  & S(K,2,2) & \cdots & S(K,m-3,2)
   & \textcolor{blue}{S(m,m-2)-f_2} &
  \textcolor{red}{S(m,m-1)} & \textcolor{red}{S(m,m)=1} \\
   S(K,\cdot,3) & S(K,1,3)  & S(K,2,3) & \cdots &
   \textcolor{blue}{S(m,m-3)-f_3}
   & \textcolor{red}{S(m,m-2)} &
  \textcolor{red}{S(m,m-1)} & \textcolor{red}{S(m,m)=1} \\
  \end{tabular}
\end{center}
where the red entries in row $s$ are Stirling numbers of the second
kind $S(m,r)$ for $m-s+1 \leq r \leq m$, and the blue entry in row $s$
is $S(m(K),m(K)-s)-f_s(K)$.

\begin{exmp}\label{exmp:chitable}
  The chromatic tables of the $2$-dimensional simplicial complexes
  from Examples~\ref{exmp:K},~\ref{exmp:MB}, and \ref{exmp:MT} are
  \begin{alignat*}{3}
    \chi(K) &=
    \begin{pmatrix}
       0 & 0 & 2 & 10 &  \textcolor{blue}{7} & \textcolor{red}{1} \\
       0 & 15 & 73 & \textcolor{blue}{62} & \textcolor{red}{15}
      &  \textcolor{red}{1}
    \end{pmatrix} & &\qquad &
    \chi(\mathrm{MB}) &=
    \begin{pmatrix}
   0  & 0 &  0 &  \textcolor{blue}{0} & \textcolor{red}{1} \\
   0 &  5 & \textcolor{blue}{20} & \textcolor{red}{10} &  \textcolor{red}{1}
    \end{pmatrix} \\
    \chi(\mathrm{MT}) &=
    \begin{pmatrix}
        0 &  0 &  0 &  0 &  0 &  \textcolor{blue}{0} &
      \textcolor{red}{1}  \\
        0 &  0 & 84 & 231 & \textcolor{blue}{126} &
       \textcolor{red}{21} &  \textcolor{red}{1}
    \end{pmatrix} & &\qquad &
    \chi(\mathrm{P}2) &=
    \begin{pmatrix}
       0 & 0 & 0 &  0 &  \textcolor{blue}{0} & \textcolor{red}{1} \\
       0 & 0 & 45 & \textcolor{blue}{55} & \textcolor{red}{15} &
       \textcolor{red}{1}
    \end{pmatrix}
     \end{alignat*}
     The red entries in column $r$ are Stirling numbers $S(m,r)$ and
     they are independent of the row index. The blue entry in row $s$
     and column $m-s$, which equals $S(m-s,s)-f_s(K)$, detects if $K$
     has maximal $s$-skeleton by Proposition~\ref{prop:S(K,m-s,s)}.
\end{exmp}

 \begin{exmp}\label{exmp:AS3}
   Let $K=\mathrm{AS}3$ be Altshuler's peculiar triangulation of the
   $3$-sphere with $f$-vector $f= (10, 45, 70, 35)$ \cite{ALT76}.  The
   $1$-chromatic polynomial is $ \chrps {\mathrm{AS}3}r1 = [r]_{10}$ as
   $K^1$ is the complete graph on $10$ vertices. The chromatic table is
   \begin{equation*}
     \chi(\mathrm{AS}3) =
     \begin{pmatrix}
           0 &    0 &    0 &    0 &    0  &   0  &   0 &    0 &
       \textcolor{blue}{0}
           &   \textcolor{red}{1} \\
    0 &    0 &    0 & 1360 & 8475 & 10355 & 4200 &  \textcolor{blue}{680} &
   \textcolor{red}{45}
   &    \textcolor{red}{1} \\
    0 &   26 &  4320 &  25915 & 38550 & 22152 & \textcolor{blue}{5845} &
    \textcolor{red}{750} &  \textcolor{red}{45}  &
    \textcolor{red}{1}
     \end{pmatrix}
   \end{equation*}
   The blue numbers determine the $f$-vector
   \begin{equation*}
     f(\mathrm{AS}3) =(10,
   S(10,9)-\chi(\mathrm{AS}3)_{19},
   S(10,8)-\chi(\mathrm{AS}3)_{28},S(10,7)-\chi(\mathrm{AS}3)_{37})
   \end{equation*}
   The row numbers of the first nonzero term in each row tell us that
   $\chs 1{\mathrm{AS}3} = 10$, $\chs 2{\mathrm{AS}3} = 4$, and $\chs
   3{\mathrm{AS}3} = 2$.

\end{exmp}

\begin{exmp}\label{exmp:LS3}
  The nonconstructible, nonshellable $3$-sphere $S^3_{17,74}$,
  $f=(17,91,148,74)$, found by Lutz \cite{lutz04}, has
\begin{gather*}
    \begin{array}{c|*{9}{c}}
   {} & r=1 & r=2 & r=3 & r=4 & r=5 & r=6 & r=7 & r=8 & r=9 \\ \hline
   s=1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 88 & 3089 \\
   s=2 & 0 & 0 & 36 & 702475 & 82949364 & 1075420155 & 3827766587 &
   5493687086 & 3876597169 \\
   s=3 & 0 & 422 & 4319865 & 338438489 & 3903094622 & 14292381565
   & 22946854806 & 19158310796 & 9202775199
  \end{array} \\
 \begin{array}{c|*{8}{c}}
  {} & r=10 & r=11 & r=12 & r=13 & r=14 & r=15 & r=16 & r=17 \\ \hline
  s=1 &  23017 & 55285 & 54973 & 25941 & 6210 & 762 & \textcolor{blue}{45}
  & \textcolor{red}{1} \\
  s=2 &  1507939074 & 346346664 & 48855523 & 4302470 & 235026
  & \textcolor{blue}{7672} &
  \textcolor{red}{136} & \textcolor{red}{1} \\
  s=3 & 2708454744 & 507528561 & 61784524 & 4903589 & \textcolor{blue}{249826}
  & \textcolor{red}{7820} &
  \textcolor{red}{136}
  & \textcolor{red}{1}
 \end{array}
\end{gather*}
as its chromatic table.  Figure~\ref{fig:SKS^3_1774} shows a
semi-logarithmic plot of the simplicial Stirling numbers
$S(S^3_{17,74},r,s)$.
The triangulation $\Sigma_{16}^3$, $f=(16, 106, 180, 90)$, of the
Poincar\'e homology $3$-sphere constructed by Bj\"orner and Lutz
\cite[Theorem 5]{bjorner-lutz00} has
 \begin{gather*}
  \begin{array}{c|*{8}{c}}
    {} & r=1 & r=2& r=3& r=4& r=5& r=6& r=7& r=8  \\ \hline
    s=1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
   s=2 & 0 & 0 & 0 & 4589 & 2974411 & 69671411 & 300475213 & 442354547 \\
s=3 & 0 & 3 & 845561 & 70005500 & 701299653 & 2158716508 & 2888730959 &
2000811501 \\
   \end{array} \\
 \begin{array}{c|*{8}{c}}
 {} & r=9& r=10& r=11& r=12& r=13& r=14& r=15 & r=16  \\ \hline
 s=1 &  0 & 0 & 0 & 0 & 28 & 44 & \textcolor{blue}{14} & \textcolor{red}{1} \\
 s=2 & 292864435 & 100793551 & 19546606 & 2225261 & 150095 &
 \textcolor{blue}{5840}  &
 \textcolor{red}{120} & \textcolor{red}{1} \\
 s=3 &  792553648 & 190527025 & 28730056 & 2750278 & \textcolor{blue}{165530} &
 \textcolor{red}{6020} & \textcolor{red}{120} & \textcolor{red}{1}
\end{array}
\end{gather*}
as its chromatic table.
\end{exmp}

\begin{figure}[t]
  \centering
  \begin{tikzpicture}
[vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  6pt, inner sep =0pt,draw=#1},
vertex/.default=black,scale=.75]

\draw[very thin, color=gray,opacity=.3] (-0.1,-0.1) grid (18.5,11.5);
\draw[->] (-0.2,0) -- (18.5,0) node[right] {$r$};
\draw[->] (0,-0.2) -- (0,11.5) node[above] {$\log_{10} S(S^3_{17,74},r,s)$};

\foreach \x in {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}
\draw (\x,-0.5) node {$\x$};

\foreach \y in {1,2,3,4,5,6,7,8,9,10,11}
\draw (-0.5,\y) node {$\y$};

\draw  plot[smooth] coordinates
{
(8, 1.9)
(9, 3.5)
(10, 4.4)
(11, 4.8)
(12, 4.8)
(13, 4.4)
(14, 3.8)
(15, 2.9)
(16, 1.7)
(17, 0.00)};
\draw (8,1.9) node[below] {$s=1$};
\foreach \x in
{(8, 1.9),
(9, 3.5),
(10, 4.4),
(11, 4.8),
(12, 4.8),
(13, 4.4),
(14, 3.8),
(15, 2.9),
(16, 1.7),
(17, 0.00)}
\draw \x node[vertex=red] {};

\draw  plot[smooth] coordinates
{(3, 1.6)
(4, 5.9)
(5, 7.9)
(6, 9.0)
(7, 9.6)
(8, 9.8)
(9, 9.6)
(10, 9.1)
(11, 8.5)
(12, 7.7)
(13, 6.6)
(14, 5.4)
(15, 3.9)
(16, 2.1)
(17, 0.00)};
\foreach \x in
{(3, 1.6),
(4, 5.9),
(5, 7.9),
(6, 9.0),
(7, 9.6),
(8, 9.8),
(9, 9.6),
(10, 9.1),
(11, 8.5),
(12, 7.7),
(13, 6.6),
(14, 5.4),
(15, 3.9),
(16, 2.1),
(17, 0.00)}
\draw \x node[vertex=red] {};
\draw (3,1.6) node[below] {$s=2$};

\draw  plot[smooth] coordinates
{(2,2.62)
(3,6.63)
(4,8.53)
(5,9.59)
(6,10.2)
(7,10.4)
(8,10.3)
(9,9.97)
(10,9.44)
(11,8.70)
(12,7.79)
(13,6.69)
(14,5.40)
(15,3.89)
(16,2.13)
(17,0.000)};
\foreach \x in
{(2,2.62),
(3,6.63),
(4,8.53),
(5,9.59),
(6,10.2),
(7,10.4),
(8,10.3),
(9,9.97),
(10,9.44),
(11,8.70),
(12,7.79),
(13,6.69),
(14,5.40),
(15,3.89),
(16,2.13),
(17,0.000)}
\draw \x node[vertex=red] {};
\draw (2,2.62) node[below] {$s=3$};
\end{tikzpicture}
  \caption{The simplicial Stirling numbers for $S^3_{17,74}$}
  \label{fig:SKS^3_1774}
\end{figure}

Observe that all the above chromatic tables have strictly
$\log$-concave rows.

 \begin{defn}\cite{sagan88}\label{defn:logconcave}
   A finite sequence $a_1,a_2,\ldots,a_N$ of $N \geq 3$ nonnegative
   integers is strictly $\log$-concave if $a_{i-1}a_{i+1} < a_i^2$ for
   $1<i<N$ (and $\log$-concave if $a_{i-1}a_{i+1} \leq a_i^2$).
 \end{defn}

 It has been conjectured that the sequence of coefficients of the
 $1$-chromatic polynomial\ of a simple graph in falling factorial form, $r \to
 S(K,1,r)$, $ \chs 1K \leq r \leq m(K)$, is $\log$-concave
 \cite[Conjecture 3.11]{brenti94}. More generally, one may ask

\begin{question}\label{question}
  Is the finite sequence of simplicial Stirling numbers
 \begin{equation*}
   r \to S(K,r,s), \qquad \chs sK \leq r \leq m(K),
 \end{equation*}
  $\log$-concave for fixed $K$ and $s$?
\end{question}

This seems to be the right question to ask as it may be true for {\em
  all\/} the chromatic polynomial s of a simplicial complex and we have seen that the
absolute value of the coefficients of the $s$-chromatic polynomial\ are simply not
$\log$-concave for $s>1$.

Note that the Stirling numbers of the second kind, which are upper
bounds for the simplicial Stirling numbers $S(K,r,s)$ by the
inequalities~\eqref{eq:bounds}, are $\log$-concave in $r$
\cite[Corollary 2]{sagan88}.

We shall now examine Question~\ref{question} on two spherical boundary
complexes of cyclic $n$-polytopes.

\begin{defn}
  $\partial\mathrm{CP}(m,n)$, $m>n$, is the $(n-1)$-dimensional simplicial
  complex on the ordered set $[m]$ with the following
  facets: An $n$-subset $\sigma$ of $[m]$ is a facet if and only if
  between any two elements of $[m]-\sigma$ there is an even number of
  vertices in $\sigma$.
\end{defn}

By Gale's Evenness Theorem \cite{gale63}, the simplicial complex
$\partial\mathrm{CP}(m,n)$ triangulates the boundary of the cyclic
$n$-polytope on $m$ vertices.  Thus $\partial\mathrm{CP}(m,n)$ is a simplicial
$(n-1)$-sphere on $m$ vertices and it is $\lfloor n/2
\rfloor$-neighborly in the sense that $\partial\mathrm{CP}(m,n)$ has the same
$s$-skeleton as the full simplex on its vertex set when $s < \lfloor
n/2 \rfloor$.

\begin{exmp}[Cyclic polytopes with $\log$-concave simplicial Stirling numbers
  of the second kind]
  Let $\partial\mathrm{CP}(m,n)$ be the triangulated boundary of the
  cyclic polytope on $m$ vertices in $\mathbf{R}^n$. The simplicial complex
  $\partial\mathrm{CP}(m,n)$ is an $m$-vertex triangulation of
  $S^{n-1}$. The chromatic tables of the simplicial $3$-spheres
  $\partial\mathrm{CP}(m,4)$ on $m=6,7,8,9,10$ vertices are
  \begin{gather*}
    \begin{pmatrix}
      0 & 0 & 0 & 0
      & \textcolor{blue}{0} & \textcolor{red}{1} \\
      0 & 1 & 21
      & \textcolor{blue}{47} & \textcolor{red}{15} & \textcolor{red}{1} \\
      0 & 16
      & \textcolor{blue}{81} & \textcolor{red}{65}
      & \textcolor{red}{15} &  \textcolor{red}{1}
    \end{pmatrix}
    \begin{pmatrix}
        0 &  0 &  0 &  0 &  0
        &  \textcolor{blue}{0} &  \textcolor{red}{1} \\
        0 &  0 & 28 & 147
        & \textcolor{blue}{112}
        & \textcolor{red}{21} &   \textcolor{red}{1} \\
        0 & 21 & 238 & \textcolor{blue}{336} & \textcolor{red}{140}
        & \textcolor{red}{21} &  \textcolor{red}{1}
    \end{pmatrix}
    \begin{pmatrix}
         0 &   0 &   0 &   0 &   0  &  0
         & \textcolor{blue}{0}   &   \textcolor{red}{1} \\
   0  &  1 &   50 & 393 &  582
   & \textcolor{blue}{226}
   &  \textcolor{red}{28}  &   \textcolor{red}{1} \\
   0  & 29 & 654 & 1533
   & \textcolor{blue}{1030} &  \textcolor{red}{266}
   &  \textcolor{red}{28} &  \textcolor{red}{1} \\
    \end{pmatrix} \\
    \begin{pmatrix}
         0 &   0 &    0 &   0 &   0 &   0 &   0
         &   \textcolor{blue}{0} &  \textcolor{red}{1} \\
   0 &   0 &  94 & 1062 & 2523 & 1719
   &  \textcolor{blue}{408}
    & \textcolor{red}{36}  &   \textcolor{red}{1} \\
   0 &  36 & 1729 & 6471 & 6591
   & \textcolor{blue}{2619} &  \textcolor{red}{462}
    & \textcolor{red}{36} &   \textcolor{red}{1}
    \end{pmatrix}
    \begin{pmatrix}
         0 &    0 &    0  &   0 &    0 &    0 &    0 &    0
         & \textcolor{blue}{0} &
         \textcolor{red}{1} \\
    0 &    1 &  180 &  2980 & 10200 & 10777 &  4225
    &  \textcolor{blue}{680}
    & \textcolor{red}{45}  &
    \textcolor{red}{1} \\
   0 &   46 &  4445 & 25960 & 38550 & 22152
   &  \textcolor{blue}{5845}
   &  \textcolor{red}{750} &   \textcolor{red}{45} & \textcolor{red}{1}
    \end{pmatrix}
  \end{gather*}
  All rows are strictly $\log$-concave. As
  $\partial\mathrm{CP}(m,4)^1=D[m]^1$, the $1$-chromatic number $\chs
  1 {\partial\mathrm{CP}(m,4)} = m$, and it is not difficult to see
  that the $2$-chromatic number $\chs 2 {\partial\mathrm{CP}(m,4)}$ is
  $2$ if $m$ is even and $3$ if $m$ is odd \cite{DMN2010}.
  \end{exmp}

Right multiplication with the upper triangular matrix $([j]_i)_{1 \leq
  i,j \leq m(K)}$ with $[j]_i = \binom ji i! = \frac{j!}{(i-j)!}$ in
row $i$ and column $j$ transforms, by Theorem~\ref{lemma:StK}, the
chromatic table into the $(\dim(K) \times m(K))$-matrix
\begin{equation*}
  \chi(K) ([j]_i)_{1 \leq i,j \leq m(K)} =
  \left( \chi^s(K,i) \right)_{\substack{1 \leq s \leq \dim(K) \\ 1
      \leq i \leq m(K)}}
\end{equation*}
with the $m(K)$ values $\chi^s(K,i)$, $1 \leq i \leq m(K)$, of the
$s$-chromatic polynomial\ in row $s$. This matrix of chromatic polynomial values
appears also to have $\log$-concave rows.

\section{Chromatic uniqueness}
\label{sec:chrunique}

In this section we briefly discuss to what extent simplicial complexes
are determined by their chromatic polynomial s.  Proposition~\label{prop:S(K,m-s,s)}
shows that the chromatic table of a simplicial complex determines its
$f$-vector.

\begin{defn}\label{defn:unique}
  $K$ is {\em chromatically unique\/} if it is determined up to isomorphism\
  by its chromatic table.
\end{defn}

In Lemma~\ref{lemma:KveeL} below, $K \amalg L$ is
the disjoint union and $K \vee L$ the one-point union of $K$ and $L$.
The proof is identical to the one for the similar statements about
chromatic polynomials for simple graphs.

\begin{lemma}\label{lemma:KveeL}
  If $K$ and $L$ are finite simplicial complexes then
  \begin{equation*}
    \chrps {K \amalg L}rs = \chrps Krs \chrps Lrs, \qquad
    \chrps {K \vee L}rs = \frac{\chrps Krs \chrps Lrs}{r}
  \end{equation*}
  for all $r$ and all $s \geq 0$.
\end{lemma}

The two nonisomorphic simplicial complexes
\begin{center}
 \begin{tikzpicture}
[vertex/.style= {shape=circle,  fill=#1!100, minimum size =
  4pt, inner sep =0pt,draw}, vertex/.default=black,scale=0.75,xscale=1.25]

\begin{scope}
    \path [fill = green!20] (0,0) -- (2,2) -- (2,-2) -- (0,0);
  \path [fill = green!20] (4,0) -- (2,2) -- (2,-2) -- (4,0);
  \path [fill = green!20] (4,0) -- (6,2) -- (6,-2) -- (4,0);

  \draw (2,2) -- (2,-2);
  \draw (0,0) -- (2,2) -- (4,0) -- (6,2) -- (6,-2) -- (4,0) -- (2,-2)
  -- (0,0) -- cycle;

  \draw (0,0) node[vertex] {};
  \draw (2,2) node[vertex] {};
  \draw (2,-2) node[vertex] {};
  \draw (4,0) node[vertex] {};
  \draw (6,-2) node[vertex] {};
  \draw (6,2) node[vertex] {};
\end{scope}

\begin{scope}[xshift=7cm]
    \path [fill = green!20] (0,0) -- (2,2) -- (2,-2) -- (0,0);
  \path [fill = green!20] (4,0) -- (2,2) -- (2,-2) -- (4,0);
  \path [fill = green!20] (4,0) -- (6,2) -- (6,-2) -- (4,0);

  \draw (0,0) -- (4,0);
  \draw (0,0) -- (2,2) -- (4,0) -- (6,2) -- (6,-2) -- (4,0) -- (2,-2)
  -- (0,0) -- cycle;

  \draw (0,0) node[vertex] {};
  \draw (2,2) node[vertex] {};
  \draw (2,-2) node[vertex] {};
  \draw (4,0) node[vertex] {};
  \draw (6,-2) node[vertex] {};
  \draw (6,2) node[vertex] {};
\end{scope}
\end{tikzpicture}
\end{center}
are not chromatically unique as they have identical chromatic tables
\begin{equation*}
  \begin{pmatrix}
     0 &  0 & 2 &10 & \textcolor{blue}{7} & \textcolor{red}{1} \\
     0  & 15 & 73
     & \textcolor{blue}{62} & \textcolor{red}{15} &  \textcolor{red}{1}
  \end{pmatrix}
\end{equation*}
by Lemma~\ref{lemma:KveeL}. (These two complexes are, however,
PL-isomorphic.)

On the other hand,
Proposition~\ref{prop:SKisSNis}.\eqref{prop:SKisSNis2} immediately
implies that the $s$-skeleton of a full simplex is chromatically
unique (in a very strong sense).

\begin{prop}
  If $K$ has the same $s$-chromatic polynomial\ as a full simplex $D[N]$, then $K$ and
  $D[N]$ have isomorphic $s$-skeleta.
\end{prop}
\begin{proof}
  If $K$ and $D[N]$ have the same $s$-chromatic polynomial\ for some $s \geq 1$, then
  $K$ has $N$ vertices (Corollary~\ref{cor:LsK}), and, since
  $\chrps K{N-s}s = \chrps {D[N]}{N-s}s$, the $s$-skeleton of $K$ is
  isomorphic to the $s$-skeleton of the full simplex on $N$ vertices
  (Proposition~\ref{prop:SKisSNis}.\eqref{prop:SKisSNis2}).
\end{proof}

\def$'${$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}

redpajama_set_name: RedPajamaArXiv


A while ago, I had blogged a Pulse article about Digital Strategy Execution (ref link). This is a topic I continue to observe and reflect on since organizations continue to execute their corporate digitization strategies.
Enabling digital strategies requires CIOs, Enterprise Architects and IT leaders to engage with business stakeholders. Such engagement of IS with business stakeholders must be governed by the organization's processes, operating model, and technology governance to ensure robust, scalable architectures. The resultant roadmaps should also be governed by a well-functioning ARB.
Most lights-on digitization efforts - like migrating application platforms to cloud hosting, introducing new software as a service (SaaS), enhanced data management, automation of existing processes etc - are driven by IT leaders, who should take the opportunity to align these with other transformations.
Technology and business leaders continually scan the external landscape for new and innovative solutions. Digital excellence initiatives include the introduction of innovative vendor solutions that can drive business growth. Examples include use of blockchain technology, tools to analyze big/unstructured data; speech recognition and interactive voice response (IVR) enabled processes, and incremental use of virtual assistants and transcription or translation services.
Customer-centric digitization programs aim to enhance digital engagement with the company's customers, business partners, vendors, suppliers, and other third parties. These initiatives require strong business insights and are generally sponsored and steered by senior executives. IT leaders facilitate ideation and technology foresight, and ensure seamless introduction of these solutions.
Customer-centric digitization might also be designed to address threats from digital innovators. For instance, the travel and hospitality industry is reacting to disruptors like Uber, Lyft, and Airbnb. Innovations in robotics, Internet of Things, and artificial intelligence–driven planning and modeling are beginning to disrupt the existing ways of working in manufacturing industries.


text:
Photos of Malta Aviation Museum
Malta Aviation Museum is an aircraft museum situated on the site of the former Royal Air Force airfield in the village of Ta'Qali, on the island of Malta. The museum, based in three hangars, covers the history of aviation on the island with exhibits, particularly from the Second World War and post-war periods. The museum is involved in the preservation and restoration of aircraft, some of which are in airworthy condition.
Continue reading Photos of Malta Aviation Museum
Format GalleryPosted on 24.10.2021 24.10.2021 Categories MaltaLeave a comment on Photos of Malta Aviation Museum
Photos of Mdina
Mdina, also known by its Italian-language titles Città Vecchia ("Old City") and Città Notabile ("Notable City"), is a fortified city in the Northern Region of Malta which served as the island's capital from antiquity to the medieval period. The city is still confined within its walls, and has a population of just under 300, but it is contiguous with the town of Rabat, which takes its name from the Arabic word for suburb, and has a population of over 11,000.
Continue reading Photos of Mdina
Format GalleryPosted on 21.10.2021 21.10.2021 Categories MaltaLeave a comment on Photos of Mdina
Photos of Marsaxlokk, Malta
Marsaxlokk is a small, traditional fishing village in the South Eastern Region of Malta. It has a harbour, and is a tourist attraction known for its views, fishermen and history. The village is also known for the Marsaxlokk Market, which is mainly a large fish market which takes place along the seafront on Sundays, and a tourist market during all other days of the week. Inhabited and well-known since antiquity, Marsaxlokk was used as a port by Phoenicians, Carthaginians and also has the remains of a Roman-era harbour.
Continue reading Photos of Marsaxlokk, Malta
Format GalleryPosted on 21.10.2021 21.10.2021 Categories MaltaLeave a comment on Photos of Marsaxlokk, Malta
Photos of Vientiane, Laos
Vientiane is the capital and largest city of Laos, on the banks of the Mekong River near the border with Thailand. Vientiane became the capital in 1563 due to fears of a Burmese invasion but was later looted then razed to the ground in 1827 by the Siamese (Thai). Vientiane was the administrative capital during French rule and, due to economic growth in recent times, is now the economic center of Laos.
Continue reading Photos of Vientiane, Laos
Format GalleryPosted on 12.03.2018 12.03.2018 Categories LaosLeave a comment on Photos of Vientiane, Laos
Video of Spain 2016
Our trip to Spain, 2016 year
Format VideoPosted on 20.08.2017 Categories SpainLeave a comment on Video of Spain 2016
Photos of Barcelona, Spain
Barcelona is the capital city of the autonomous community of Catalonia in the Kingdom of Spain, as well as the country's second most populous municipality, with a population of 1.65 million within city limits. Its urban area extends beyond the administrative city limits with a population of around 4.7 million people, being the seventh-most populous urban area in the European Union after Paris, London, Madrid, the Ruhr area, Berlin, and Milan. It is the largest metropolis on the Mediterranean Sea.
Continue reading Photos of Barcelona, Spain
Format GalleryPosted on 02.08.2016 Categories SpainLeave a comment on Photos of Barcelona, Spain
Photos of Spanish province
All pictures on this page was created in June-July 2016.
Continue reading Photos of Spanish province
Format GalleryPosted on 02.08.2016 02.08.2016 Categories SpainLeave a comment on Photos of Spanish province
Photos of Ciudad Encantada, Spain
The Ciudad Encantada (Enchanted City) is a geological site near the city of Cuenca, in the autonomous community of Castilla-La Mancha, Spain in which the erosive forces of weather and the waters of the nearby Júcar river have formed rocks into distinctive and memorable shapes. It was declared a Natural Site of National Interest on 11 June 1929. The rock formations of Ciudad Encantada are karst formations, which date back to the Cretaceous period, approximately 90 million years ago.
Continue reading Photos of Ciudad Encantada, Spain
Format GalleryPosted on 29.07.2016 Categories SpainLeave a comment on Photos of Ciudad Encantada, Spain
Photos of Cabo de Gata-Nijar Natural Park, Spain
Cabo de Gata-Nijar Natural Park in the south-eastern corner of Spain, is Andalucia's largest coastal protected area, a wild and isolated landscape with some of Europe's most original geological features. It is the only region in Europe with a true hot desert climate. The eponymous mountain range of the Sierra del Cabo de Gata with its highest peak El Fraile is Spain's largest volcanic rock formation with sharp peaks and crags in red/ochre-hues. It falls steeply to the Mediterranean Sea creating jagged 100-metre high cliffs, which are riven by gullies leading to hidden coves with white sandy beaches, some of the most beautiful in Andalucia.
Continue reading Photos of Cabo de Gata-Nijar Natural Park, Spain
Format GalleryPosted on 29.07.2016 Categories SpainLeave a comment on Photos of Cabo de Gata-Nijar Natural Park, Spain
Photos of Morella, Spain
Morella is an ancient walled city located on a hill-top in the province of Castellón, Valencian Community, Spain. The town is the capital and administrative centre of the comarca of Els Ports, in the historic Maestrat (Maestrazgo) region. There are traces of settlement by the Iberians, succeeded by the Greeks and Romans, Visigoths and the Moors. From the early 17th century to the Spanish Civil War, the town was often fought over, due to its strategic situation between the Ebro and the coastal plain of Valencia.
Continue reading Photos of Morella, Spain
Format GalleryPosted on 27.07.2016 Categories SpainLeave a comment on Photos of Morella, Spain

redpajama_set_name: RedPajamaCommonCrawl


DULLES, Va.Oct. 4, 2007--Orbital Sciences Corporation (NYSE:ORB), the world leader in smaller-sized commercial geosynchronous (GEO) communications satellites, is in final preparations for the launch of two of its spacecraft from the European space launch complex in French Guiana. The two satellites are the Intelsat-11 (IS-11) spacecraft for Intelsat, Ltd. of Bermuda and the Optus D2 satellite for Optus Networks, Pty of Australia. The mission is currently scheduled for Friday, October 5, with the launch to take place within an available window that extends from 5:28 p.m. to 6:13 p.m. (EDT).
In another sign of Orbital's rapidly growing GEO satellite business, two of Orbital's Star(TM) satellites will be launched aboard the same rocket for the first time. In previous launches aboard Ariane rockets, Orbital's smaller-sized satellites have been co-manifested with another larger satellite from a different manufacturer. Thus far in 2007, Orbital has been awarded five firm and four optional orders for its GEO satellites, making it the most productive year in the company's history for commercial communications satellites.
"Our team is very excited to be in the final phase of preparing the IS-11 and Optus D2 spacecraft for their launch and deployment, which will be followed by several weeks of in-orbit check-out before they are maneuvered into their operational slots 22,300 miles above the Earth," said Senior Vice President Christopher Richmond, head of Orbital's GEO communications satellite programs. "Both Intelsat and Optus are very important and valued customers of Orbital and we are committed to extending our history of carrying out successful missions for them."
Following its launch and check-out, the IS-11 spacecraft will join several other Orbital-built satellites in Intelsat's in-orbit fleet, including Galaxy 12, 14 and 15. The IS-11 spacecraft is a hybrid satellite that will provide both C-band and Ku-band services. It will deliver C-band services to Intelsat's customers in the continental United States and will also serve as DirecTV Latin America's Ku-band downlink for coverage of Brazil. In addition to IS-11, Intelsat has three other Orbital GEO spacecraft on order, including Horizons-2 (for a joint venture between Intelsat and JSAT of Japan), and Intelsat-15 and Intelsat-16.
Built for Australia-based Optus Networks, Pty., Optus D2 is the second in a series of satellites that will provide Ku-band fixed communications and direct television broadcasting services to Australia and New Zealand. The Optus D2 is among the most powerful GEO communications satellites ever built by Orbital, designed to generate 5 Kw of electrical power. Optus D2 will carry 24 active Ku-band transponders on a platform that is ideal for telephony, data and broadcasting applications. Earlier this year, Optus ordered its third satellite from Orbital for the D-series of spacecraft. Orbital is scheduled to deliver the Optus D3 satellite in 2009.
Orbital's commercial satellite business has experienced rapid growth over the past several years. Including the five new orders the company has booked so far this year, Orbital currently has 10 commercial GEO satellites and five science and defense spacecraft in various stages of design, production and testing at its Dulles, VA satellite manufacturing facility. To accommodate this rapid rate of growth, earlier this year Orbital completed an expansion of its manufacturing plant that, together with other facility improvements, has increased its manufacturing throughput capacity by over 30% as compared to one year ago.


Lincolnville Family Dentistry, dentist, listed under "Dentists" category, is located at Rr 1 Lincolnville ME, 04849 and can be reached by 2077895270 phone number. Lincolnville Family Dentistry has currently 0 reviews.
Browse all Dentists in Lincolnville ME. Discover census data for Lincolnville, ME.