text:
\section{Introduction}\label{s:background}

The maximally clustered permutations introduced in \cite{losonczy} are a
generalization of the freely braided permutations developed in \cite{g-l1} and
\cite{g-l2}, and these in turn include the fully commutative permutations
studied in \cite{s1} as a subset.  In \cite{j2}, an explicit formula was
obtained for the Kazhdan--Lusztig polynomials of maximally-clustered
hexagon-avoiding permutations, generalizing an earlier result of \cite{b-w}
that identified the 321-hexagon avoiding permutations.  The enumeration of the
321-hexagon avoiding permutations was first given by \cite{s-w} who showed
that these elements satisfy a linear constant-coefficient recurrence with 7
terms.

\begin{theorem}{\bf \cite{s-w}}\label{t:s-w}
The number $c_n$ of 321-hexagon-avoiding permutations in $S_n$ satisfies the
recurrence
\[ c_{n+1} = 6 c_{n} - 11c_{n-1} + 9c_{n-2} - 4c_{n-3} - 4c_{n-4} + c_{n-5} \]
for all $n \geq 8$ with initial conditions given in Figure~\ref{f:mc.enum}.
\end{theorem}

Theorem~\ref{t:s-w} was extended in \cite{mansour-stankova} and also proved
using an enumeration scheme as described in \cite{vatter}.
Figure~\ref{f:mc.enum} shows the number $b_n$ and $m_n$ of freely-braided
hexagon-avoiding and maximally-clustered hexagon-avoiding permutations
respectively, in $S_n$ for $n \leq 15$.  It can be shown that each of these
pattern classes have a rational generating function by Proposition 13 and
Corollary 10 of \cite{a-l-r}.
In this paper, we develop a method for determining the generating function
precisely.

{\tiny
\begin{figure}[h]
\begin{center}
\begin{tabular}{|p{1.05in}|p{0.1in}|p{0.15in}|p{0.15in}|p{.15in}|p{0.15in}|p{0.2in}|p{0.2in}|p{0.25in}|p{0.25in}|p{0.3in}|p{.35in}|p{0.4in}|p{0.4in}|p{0.4in}|}
\hline
$n$ & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\
\hline
\# $[321]$-hexagon avoiding & 5 & 14 & 42 & 132 & 429 & 1426 & 4806 & 16329 & 55740 & 190787 & 654044 & 2244153 & 7704047 \\
\hline
\# freely-braided hexagon-avoiding & 6 & 20 & 71 & 260 & 971 & 3670 & 13968 & 53369  & 204352  & 783408  & 3005284 &  11533014 &  44267854 \\
\hline
\# maximally-clustered hexagon-avoiding & 6 & 21 & 78 & 298 & 1157 & 4535 & 17872 & 70644 & 279704  & 1108462 & 4395045 & 17431206 &  69144643 \\
\hline
\end{tabular}
\end{center}
\caption{Enumeration of hexagon-avoiding classes}
\label{f:mc.enum}
\end{figure}
}

One feature of our main results, Theorem~\ref{t:enumerate} and
Theorem~\ref{t:main_diamond} below, is that they provide many examples of
classes characterized by permutation pattern avoidance that have rational
generating functions.  We also apply Theorem~\ref{t:enumerate} to enumerate the
maximally-clustered permutations introduced in \cite{losonczy}.  The freely
braided permutations have previously been enumerated in \cite{mansour} and we
recover this result from Theorem~\ref{t:enumerate} as well.

Section~\ref{s:background} describes the combinatorial model used in our
enumeration.  In Section~\ref{s:enumeration} we give a result that enables us
to find enumerative formulas for the pattern classes mentioned above in a
unified way.  In Section~\ref{s:diamond} we show how the methods of proof from
Section~\ref{s:enumeration} can be applied in the fully commutative case to
recover the main result of \cite{s-w} as well as some new generating functions.

\subsection{Background}

We view the symmetric group $S_n$ as the Coxeter group of type $A$ with
generating set $S = \{ s_1, \dots, s_{n-1} \}$ and relations of the form $(s_i
s_{i \pm 1})^3 = 1$ together with $(s_{i}s_{j})^{2} = 1$ for $| i - j | \geq
2$ and ${s_i}^2 = 1$.  We also refer to elements in the symmetric group by the \textit{1-line
notation} $w=[w_{1}w_{2} \dots w_{n}]$ where $w$ is the bijection mapping $i$
to $w_{i}$.  Then the generators $s_{i}$ are the adjacent transpositions
interchanging the entries $i$ and $i+1$ in the 1-line notation.  Suppose
$w=[w_1 \dots w_n]$ and $p=[p_1 \ldots p_k]$ is another permutation in $S_{k}$
for $k\leq n$.  We say $w$ \textit{contains the permutation pattern} $p$ or $w$
\textit{contains} $p$ \textit{as a 1-line pattern} whenever there exists a
subsequence $1\leq i_{1}<i_{2}<\ldots<i_{k}\leq n$ such that
\begin{equation*}
w_{i_{a}} < w_{i_{b}} \text{ if and only if } p_{a} < p_{b}
\end{equation*}
for all $1 \leq a < b \leq k$.  We call $(i_1, i_2, \dots, i_k)$ the \em
pattern instance\em.  For example, $[\ul{5}32\ul{4}\ul{1}]$ contains the
pattern $[321]$ in several ways, including the underlined subsequence.  If $w$
does not contain the pattern $p$, we say that $w$ \em avoids \em $p$.
A \em pattern class \em is a set of permutations characterized by avoiding a set
of permutation patterns.  For example, the maximally clustered permutations are
characterized by avoiding the permutation patterns
\begin{align}\label{e:mc.patterns}
    \text{ $[3421]$, $[4312]$, and $[4321]$ }
\end{align}
by \cite[Proposition 3.7]{losonczy},
while the freely braided permutations are characterized by avoiding
\begin{align}\label{e:fb.patterns}
    \text{ $[4231]$, $[3421]$, $[4312]$, and $[4321]$ }
\end{align}
as permutation patterns by \cite[Proposition 5.1.1]{g-l1}.

Recall that the products of generators from $S$ with a minimal number of
factors are called \em reduced expressions\em, and $l(w)$ is the length of such
an expression for $w \in S_n$.  Given $w \in S_n$, we represent reduced
expressions for $w$ in sans serif font, say $\mathsf{w}=\mathsf{w}_{1} \mathsf{w}_{2}\cdots \mathsf{w}_{p}$
where each $\mathsf{w}_{i} \in S$.  We call any expression of the form $s_i s_{i \pm 1}
s_i$ a \em short-braid\em.  There is a well-known theorem of Matsumoto
\cite{matsumoto} and Tits \cite{t}, which states that any reduced expression for
$w$ can be transformed into any other by applying a sequence of relations of the
form $(s_i s_{i \pm 1})^3 = 1$ together with $(s_{i}s_{j})^{2} = 1$ for $| i - j
| > 1$.  The theorem implies that the set of all generators appearing in any
reduced expression for $w$ is well-defined.  We call this set of generators the
\em support \em of $w$ and denote it by $supp(w)$.  We say that the permutation
$w$ is \em connected \em if the subscripts of the generators appearing in
$supp(w)$ form a nonempty interval in $\{1, 2, \ldots, n-1\}$.

As in \cite{s1}, we define an equivalence relation on the set of reduced
expressions for a permutation by saying that two reduced expressions are in the
same \em commutativity class \em if one can be obtained from the other by a
sequence of \em commuting moves \em of the form $s_i s_j \mapsto s_j s_i$ where
$|i-j| \geq 2$.  If the reduced expressions for a permutation $w$ form a single
commutativity class, then we say $w$ is \em fully commutative\em.

\subsection{Heaps}

If $\mathsf{w} = \mathsf{w}_1 \cdots \mathsf{w}_k$ is a reduced expression, then as in \cite{s1} we
define a partial ordering on the indices $\{1, \cdots, k\}$ by the transitive
closure of the relation $i \prec j$ if $i < j$ and $\mathsf{w}_i$ does not commute with
$\mathsf{w}_j$.  We label each element $i$ of the poset by the corresponding generator
$\mathsf{w}_i$.  It follows from the definition that if $\mathsf{w}$ and $\mathsf{w}'$ are two reduced
expressions for a permutation $w$ that are in the same commutativity class, then
the labeled posets of $\mathsf{w}$ and $\mathsf{w}'$ are isomorphic.  This isomorphism class of
labeled posets is called the \em heap \em of $\mathsf{w}$, where $\mathsf{w}$ is a reduced
expression representative for a commutativity class of $w$.  In particular, if
$w$ is fully commutative then it has a single commutativity class, and so there
is a unique heap of $w$.

As in \cite{b-w}, we will represent a heap as a set of lattice points embedded
in $\N^2$.  To do this, we assign coordinates $(x,y) \in \N^2$ to each entry of
the labeled Hasse diagram for the heap of $\mathsf{w}$ in such a way that:
\begin{enumerate}
\item[(1)] An entry represented by $(x,y)$ is labeled $s_i$ in the heap if and
only if $x = i$, and
\item[(2)] If an entry represented by $(x,y)$ is greater than an entry
represented by $(x',y')$ in the heap, then $y > y'$.
\end{enumerate}
Since the Coxeter graph of type $A$ is a path, it follows from the definition
that $(x,y)$ covers $(x',y')$ in the heap if and only if $x = x' \pm 1$, $y >
y'$, and there are no entries $(x'', y'')$ such that $x'' \in \{x, x'\}$ and $y'
< y'' < y$.  Hence, we can completely reconstruct the edges of the Hasse
diagram and the corresponding heap poset from a lattice point representation.
This representation will enable us to make arguments ``by picture'' that would
otherwise be difficult to formulate.  Although there are many coordinate
assignments for any particular heap, the $x$ coordinates of each entry are
fixed for all of them, and the coordinate assignments of any two entries only
differs in the amount of vertical space between them.

\begin{example}\label{ex:heap}
One lattice point representation of the heap of $w = s_2 s_3 s_1 s_2 s_4$ is
shown below, together with the labeled Hasse diagram for the unique heap poset
of $w$.

\smallskip
\begin{center}
\begin{tabular}{ll}
\xymatrix @=-4pt @! {
\hs & \hs & \hs & \hs & \hs \\
& \hf & \hs & \hf & \hs & \hs \\
\hf & \hs & \hf & \hs & \hs & \hs \\
& \hf & \hs & \hs & \hs & \hs \\
s_1 & s_2 & s_3 & s_4 \\
} &
\xymatrix @=0pt @! {
 & \hf_{s_2} \ar@{-}[dl] \ar@{-}[dr] & & \hf_{s_4} \ar@{-}[dl] \\
 \hf_{s_1} \ar@{-}[dr] & & \hf_{s_3} \ar@{-}[dl] \\
 & \hf_{s_2} \\
} \\
\end{tabular}
\end{center}
\end{example}

Suppose $x$ and $y$ are a pair of entries in the heap of $\mathsf{w}$ that correspond
to the same generator $s_i$, so that they lie in the same column $i$ of the heap.
Assume that $x$ and $y$ are a \em minimal pair \em in the sense that there is
no other entry between them in column $i$.  Then, for $\mathsf{w}$ to be reduced, there
must exist at least one non-commuting generator between $x$ and $y$, and if
$\mathsf{w}$ is short-braid avoiding, there must actually be two non-commuting labeled
heap entries that lie strictly between $x$ and $y$ in the heap.  We call these
two non-commuting labeled heap entries a \em resolution \em of the pair $x,y$.
If the generators lie in distinct columns, we call the resolution a \em distinct
resolution\em.  The Lateral Convexity Lemma of \cite{b-w} characterizes
fully commutative permutations $w$ as those for which every minimal pair in the
heap of $w$ has a distinct resolution.

\begin{definition} \label{d:supports}
If $s_i \in supp(w)$, we say that $s_i$ \emph{supports} $w$.  We also
say that column $i$ supports $w$.  If $w$ is fully commutative and
connected then $supp(w) = \{s_i, s_{i+1}, \ldots, s_j\}$
for some $i, j$, and since every minimal pair of entries in the heap of $w$ has
a distinct resolution we must have exactly one entry in columns $i$ and $j$ of
the heap.  In this situation, we call columns $i+1, i+2, \ldots, j-1$ the \em
internal columns \em of the heap of $w$ and we call columns $i$ and $j$ the \em
extremal columns \em of the heap of $w$.
\end{definition}

We now describe a notion of containment for heaps.  Recall from \cite{b-j1} that
an \em orientation preserving Coxeter embedding \em $f: \{s_1, \dots, s_{k-1} \}
\rightarrow \{s_1, \dots, s_{n-1} \}$ is an injective map of Coxeter generators such
that for each $m \in \{2, 3\}$, we have
\[ (s_i s_j)^{m} = 1 \text{ if and only if } (f(s_i) f(s_j))^{m} = 1 \]
and the subscript of $f(s_i)$ is less than the subscript of $f(s_j)$ whenever $i
< j$.  We view this as a map of permutations, which we also denote $f: S_k
\rightarrow S_n$, by extending it to a word homomorphism which can then be
applied to any reduced expression in $S_k$.

Recall that a subposet $Q$ of $P$ is called \em convex \em if $y \in Q$
whenever $x < y < z$ in $P$ and $x, z \in Q$.  Suppose that $w$ and $h$ are
permutations.  We say that $w$ \em heap-contains \em $h$ if there exist
commutativity classes represented by $\mathsf{w}$ and $\mathsf{h}$, together with an orientation
preserving Coxeter embedding $f$ such that the heap of $f(\mathsf{h})$ is contained as
a convex labeled subposet of the heap of $\mathsf{w}$.  If $w$ does not heap-contain
$h$, we say that $w$ \em heap-avoids \em $h$.  To illustrate, $w = s_2 s_3 s_1
s_2 s_4$ from Example~\ref{ex:heap} heap-contains $s_1 s_2 s_3$ under the
Coxeter embedding that sends $s_i \mapsto s_{i+1}$, but $w$ heap-avoids $s_1
s_2 s_1$.

In type $A$, the heap construction can be combined with another
combinatorial model for permutations in which the entries from the
1-line notation are represented by strings.  The points at which two
strings cross can be viewed as adjacent transpositions of the 1-line
notation.  Hence, we may overlay strings on top of a heap diagram to
recover the 1-line notation for the permutation, by drawing the strings
from bottom to top so that they cross at each entry in the
heap where they meet and bounce at each lattice point not in
the heap.  Conversely, each permutation string diagram corresponds
with a heap by taking all of the points where the strings cross as the
entries of the heap.

For example, we can overlay strings on the two heaps of $[3214]$.
Note that the labels in the picture below refer to the strings, not
the generators.
\begin{center}
\begin{tabular}{ll}
    \heap{
    & 3 \ \ 2 &  & 1 \ \ 4 & \\
    \StringR{\hs} & \hs & \StringLRX{\hf} & \hs & \StringL{\hs} \\
    \hs & \StringLRX{\hf} & \hs & \StringLR{\hs} & \hs \\
    \StringR{\hs} & \hs & \StringLRX{\hf} & \hs & \StringL{\hs} \\
    & 1 \ \ 2 &  & 3 \ \ 4 & \\
    } &
    \heap{
    & 3 \ \ 2 &  & 1 \ \ 4 &  \\
    \hs & \StringLRX{\hf} & \hs & \StringLR{\hs} \\
    \StringR{\hs} & \hs & \StringLRX{\hf} & \hs & \StringL{\hs} \\
    \hs & \StringLRX{\hf} & \hs & \StringLR{\hs} \\
    & 1 \ \ 2 &  & 3 \ \ 4 &  \\
    } \\
\end{tabular}
\end{center}

For a more leisurely introduction to heaps and string diagrams, as well as
generalizations to Coxeter types $B$ and $D$, see \cite{b-j1}.  Cartier and
Foata \cite{cartier-foata} were among the first to study heaps of dimers, which
were generalized to other settings by Viennot \cite{viennot}.  Stembridge has
studied enumerative aspects of heaps \cite{s1,s2} in the context of fully
commutative elements.  Green has also considered heaps of pieces with
applications to Coxeter groups in \cite{green1,green2,green3}.

\subsection{Maximally clustered elements}

In \cite{losonczy}, Losonczy introduced the maximally clustered elements of
simply laced Coxeter groups.

\begin{definition}{\bf \cite{losonczy}}\label{d:mc}
A \em braid cluster \em is an expression of the form \[ s_{i_1} s_{i_2} \dots
s_{i_k} s_{i_{k+1}} s_{i_k} \dots s_{i_2} s_{i_1} \] where each $s_{i_p}$ for
$1 \leq p \leq k$ has a unique $s_{i_q}$ with $p < q \leq k+1$ such that
$|i_p - i_q| = 1$.

Let $w$ be a permutation and let $N(w)$ denote the number of $[321]$ pattern
instances in $w$.  We say $w$ is \em maximally clustered \em if there is a
reduced expression for $w$ of the form
\[ a_0 c_1 a_1 c_2 a_2 \dots c_M a_M \]
where each $a_i$ is a reduced expression, each $c_i$ is a braid cluster with
length $2 n_i + 1$ and $N(w) = \sum_{i=1}^{M} n_i$.  Such an expression is
called \em contracted\em.  In particular, $w$ is \em freely braided \em if
there is a reduced expression for $w$ with $N(w)$ disjoint short-braids.
\end{definition}

This is not the original definition for the maximally clustered elements;
however it is equivalent.  The remarks in Section 5 of \cite{g-l1} show that the
number of $[321]$ pattern instances in $w$ equals the number of contractible
triples of roots in the inversion set of $w$.  Corollary 4.11(ii) and
Corollary 4.13 of \cite{losonczy} prove that $\mathsf{w}$ is a contracted reduced
expression for a maximally clustered element if and only if it has the form
given in Definition~\ref{d:mc}.  Moreover, it follows from the proof of
\cite[Corollary 4.11(ii)]{losonczy} that the $a_i$ in Definition~\ref{d:mc} are
fully commutative.

Recall that \cite{b-j-s} showed that $w$ is fully commutative whenever $N(w) =
0$.  In this work we will frequently use the fact that any braid cluster has the
canonical form of Lemma~\ref{l:braid_cluster}.

\begin{lemma}\label{l:braid_cluster}
Suppose $x = s_{i_1} s_{i_2} \dots s_{i_{k}} s_{i_{k+1}} s_{i_{k}} \dots s_{i_2}
s_{i_1}$ is a braid cluster of length $2 k + 1$ in type $A$.  Then, $x = s_{m+1}
s_{m+2} \dots s_{m+{k}} s_{m+k+1} s_{m+{k}} \dots s_{m+2} s_{m+1}$ for some $m$.
\end{lemma}
\begin{proof}
Since $s_{i_{k+1}}$ is a transposition and conjugation preserves cycle type, $x$
is a transposition and we may write $x = (m+1 \,\,\,\, m+k'+2)$ in cycle
notation for some $k',m \geq 0$.  This transposition is given by the expression
$\mathsf{x} = s_{m+1} s_{m+2} \cdots s_{m+k'} s_{m+k'+1} s_{m+k'} \cdots s_{m+2}
s_{m+1}$, which is a reduced expression for $x$ by \cite[Lemma 4.3]{losonczy}.
Since the length of reduced expressions for $x$ is an invariant of $x$, $k' = k$
and the result follows.
\end{proof}

Recall the following structural lemma about contracted reduced expressions.

\begin{lemma}{\bf \cite[Lemma 2.3]{j2}}\label{l:unique}
Let $\mathsf{w}$ be a contracted reduced expression for a maximally clustered
permutation, so $\mathsf{w}$ has the form
\[ a_0 c_1 a_1 \dots c_{M} a_{M} \]
where each $c_j$ is a braid cluster, and the $a_j$ are short-braid
avoiding.  Then, any generator $s_i$ that appears in any of the braid clusters
$c_j$ does not appear anywhere else in $\mathsf{w}$.
\end{lemma}

\begin{lemma}\label{l:constant_support}
Let $\mathsf{w}$ be a contracted reduced expression for a maximally clustered
permutation.  If the generator $s_i$ supports a braid cluster in $\mathsf{w}$ then $s_i$
supports a braid cluster in every contracted reduced expression for $w$.
\end{lemma}
\begin{proof}
By \cite[Lemma 4.11(i)]{losonczy}, the contracted reduced expressions form a
complete set of representatives for the commutativity classes of $w$.  Suppose
the generator $s_i$ supports a braid cluster in $\mathsf{w}$ in the sense of
Definition~\ref{d:supports} but there exists a contracted reduced expression
$\mathsf{w}' = a_0 c_1 a_1 \ldots c_M a_M$ for $w$ in which $s_i$ does not support a
braid cluster.  By the theorem of Matsumoto \cite{matsumoto} and Tits \cite{t},
it suffices to consider a pair of heaps represented by $\mathsf{w}$ and $\mathsf{w}'$ that are
related by a single short-braid move.

Observe that each short-braid move on $\mathsf{w}'$ can change the length of at most
one braid cluster.  This is clear if the short-braid move involves two entries
from a single braid cluster, or one entry from some braid cluster $c_i$
together with some entry of $a_j$ for $j \in \{i-1, i\}$.  In the case that
some $a_i$ is the identity, note that there are no short-braid moves involving
an entry from $c_i$ and an entry from $c_{i+1}$ by Lemma~\ref{l:unique}.

Therefore, by the equation $N(w) = \sum_{i = 1}^{M} n_i$ from
Definition~\ref{d:mc}, we actually have that no short-braid move on $\mathsf{w}'$ can
change the length of any of the braid clusters because $N(w)$ does not depend on
the reduced expression for $w$.  Thus we have shown that the length of each
braid cluster remains the same over all contracted reduced expressions for $w$.
Finally, since no single short-braid move can change the support of a
braid-cluster without changing its length, we have shown that the support of
each braid cluster also remains the same over all contracted reduced expressions
for $w$.
\end{proof}

Putting these lemmas together, we can show that there is a canonical heap
associated to any maximally clustered permutation.  See
Example~\ref{ex:canonical} for an illustration.

\begin{definition}\label{d:bc_column_decomposition}
Let $\mathfrak{C}$ be a commutativity class for a permutation $w$.
Suppose that the set of columns $\{1, \ldots, n\}$ of the heap of $\mathfrak{C}$ can be
partitioned into intervals
\[ \tilde{C}_0 = [1, p_1 -1], \tilde{B}_1 = [p_1, q_1], \tilde{C}_1 = [q_1+1,
p_2-1], \ldots, \tilde{B}_k = [p_k, q_k], \tilde{C}_k = [q_k + 1, n] \]
satisfying:
\begin{enumerate}
\item[(1)]  For each $i$, the heap of $\mathfrak{C}$ restricted to
$\tilde{B}_i$ is a braid cluster in the canonical form of
Lemma~\ref{l:braid_cluster}.
\item[(2)]  For each $i$, every minimal pair of entries in any column from
$\tilde{C}_i$ has a distinct resolution, possibly using entries from columns
$q_i$ or $p_{i+1}$.
\end{enumerate}
Then, we say that the heap of $\mathfrak{C}$ has a \em braid cluster column
decomposition \em given by $\tilde{C}_0, \tilde{B}_1, \tilde{C}_1, \ldots
\tilde{B}_k, \tilde{C}_k$.
\end{definition}

\begin{proposition}\label{p:canonical_form}
Every maximally clustered permutation $w$ has a unique commutativity class
$\mathfrak{C}$ such that the heap of $\mathfrak{C}$ has a braid cluster
column decomposition.  Conversely, every heap having a braid cluster column
decomposition corresponds to a maximally clustered permutation.
\end{proposition}
\begin{proof}
Let $w$ be a maximally clustered permutation.  By \cite[Lemma
4.11(i)]{losonczy}, the contracted reduced expressions form a complete set of
representatives for the commutativity classes of $w$.  By
Lemma~\ref{l:constant_support}, each generator either supports a unique braid
cluster in some contracted reduced expression, in which case the generator must
also support the braid cluster in every contracted reduced expression, or the
generator is not used in any braid cluster.  Hence, the set of braid clusters
appearing in any contracted reduced expression for $w$ is independent of the
choice of reduced expression.  Therefore, specifying a commutativity class of
$w$ depends only on a choice of commutativity class for each braid cluster.  We
choose each braid cluster to have the form given in Lemma~\ref{l:braid_cluster}
and declare this to be our canonical commutativity class $\mathfrak{C}$ for $w$.

Consider the heap associated to $\mathfrak{C}$.  We can partition the support of
$w$ into segments $\tilde{B}_1, \tilde{B}_2, \ldots, \tilde{B}_k$ corresponding
to the generators that support braid clusters.  The complementary columns $\{1,
\ldots, n\} \setminus \bigcup_{i=1}^k \tilde{B}_i$ form intervals that we denote
$\tilde{C}_0, \tilde{C}_1, \ldots, \tilde{C}_k$.

By definition, this column partition satisfies property (1) of
Definition~\ref{d:bc_column_decomposition}.  Any minimal pair of entries from
some $\tilde{C}_i$ without a distinct resolution would either form a
short-braid or form an extension of an existing braid-cluster, contradicting
the definition of the column partition we have given.  Hence, the column
partition satisfies property (2) of
Definition~\ref{d:bc_column_decomposition} as well.

To prove the converse statement, fix a heap with a column partition satisfying
properties (1) and (2) and let $\mathsf{w}$ be a corresponding expression.  We show that
$\mathsf{w}$ is reduced and maximally clustered.

If $\mathsf{w}$ is not reduced then there exists a sequence of braid moves bringing $\mathsf{w}$
to an expression whose heap has two entries in the same column with no entries
between them.  But the available braid moves in any expression related to $\mathsf{w}$
only involve entries from some braid cluster supported on columns
$\tilde{B}_i$ because the entries in the complimentary columns $\bigcup_{i = 0}^k
\tilde{C}_i$ all have distinct resolutions by property (2) and this remains true
after performing arbitrarily many braid moves in $\bigcup_{i=1}^k \tilde{B}_i$.
Since the braid clusters supported on each $\tilde{B}_i$ are reduced, we never
have any opportunity for cancellation.  Hence, $\mathsf{w}$ is reduced.

Moreover, we can read a contracted reduced expression for $w$ from the given
heap.  We begin by linearly ordering the braid clusters
$\tilde{B}_1, \dots, \tilde{B}_k$ in a way which is compatible with the
order in which they appear in the heap poset.  First read all of the entries in
columns $\tilde{C} = \bigcup_{i = 0}^k \tilde{C}_{i}$
that lie below the entries of $\tilde{B}_1$ in the heap poset.  Then for
each $i < k$ we sequentially read the entries of $\tilde{B}_i$ followed by
the entries from columns $\tilde{C}$ appearing between $\tilde{B}_i$ and
$\tilde{B}_{i+1}$ in the heap poset.  Finally, we read the entries from
columns $\tilde{C}$ that appear above $\tilde{B}_{k}$ in the heap poset.

\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\heap {
& & \StringLRX{\hb} & & \StringLR{\hs} & & \StringR{\hs}\\
& \StringR{\hs} & & \StringLRX{\hb} & & \StringL{\hs} & & \StringL{\hs}\\
& & \StringLR{\hs} & & \StringLRX{\hb} & & \StringRXD{\hf}\\
& \StringR{\hs} & & \StringLR{\hs} & & \StringLRX{\hb} & & {\hf} \\
& & \StringLR{\hs} & & \StringLRX{\hb} & & \StringLXD{\hf}\\
& \StringR{\hs} & & \StringLRX{\hb} & & \StringL{\hs} & & \StringL{\hs}\\
& & \StringLRX{\hb} & & \StringLR{\hs} & & \StringR{\hs}\\
& \StringRX{\hf} & & \StringLR{\hs} & & \StringL{\hs} & & \StringL{\hs}\\
} \\
\end{tabular}
\end{center}
\caption{Strings on a braid cluster}
\label{f:overlay}
\end{figure}

Since the braid clusters have the canonical form of Lemma~\ref{l:braid_cluster},
we can verify that each braid cluster $\tilde{B}_i$ supported on $n_i + 1$
columns has $n_i$ $[321]$-instances using a string diagram overlaid on the heap,
as illustrated in Figure~\ref{f:overlay}.  By Lemma~\ref{l:unique} each braid
cluster $\tilde{B}_{i}$ takes exactly one string lying next to a column in
$\tilde{C}_{i}$ and crosses it with exactly one string lying next to a column in
$\tilde{C}_{i+1}$.  Since each minimal pair of entries from any column of
$\tilde{C}_{i}$ has a distinct resolution by property (2), there are no other
$[321]$-instances in $w$.  Therefore, the number of $[321]$-instances in $w$ is
exactly $\sum_{i=0}^{k-1} n_i$, and each braid cluster $\tilde{B}_{i}$ has
length $2 n_i + 1$.  Hence, $w$ is maximally clustered by Definition~\ref{d:mc}.
\end{proof}

\begin{example}\label{ex:canonical}
Suppose $w$ is given by the contracted reduced expression
\[ (s_5) (s_1 s_2 s_3 s_4 s_3 s_2 s_1) (s_6 s_5 s_9) (s_7 s_8 s_7) (s_6).\]
Then the heap of $w$ is drawn below, together with its braid cluster column
decomposition.  The braid clusters are shown in grey.
{\scriptsize
\[ \xymatrix @=-8pt @! {
& & {\hs} & & {\hs} & & {\hf} & & {\hs} & \\
& {\hb} & & {\hs} & & {\hs} & & {\hb} \\
& & {\hb} & & {\hs} & & {\hs} & & {\hb} & \\
& {\hs} & & {\hb} & & {\hf} & & {\hb} & & {\hf} \\
& & {\hs} & & {\hb} & & {\hf} & & {\hs} & \\
& {\hs} & & {\hb} & & {\hf} & & {\hs} & \\
& & {\hb} & & {\hs} & & {\hs} & & {\hs} \\
& {\hb} & & {\hs} & & {\hs} & & {\hs} & \\
& & {\hs} & & {\hs} & & {\hs} & & {\hs} \\
& \text{[} \tilde{B}_1 & & & \text{]} & \text{[} \tilde{C}_1 & \text{]} &
\text{[} \tilde{B}_2  & \text{]} & \text{[} \tilde{C}_3 \text{]} \\ } \] }
\end{example}

\bigskip
\section{Enumeration}\label{s:enumeration}

Let $S^{P} = \bigcup_{n \geq 1} S^{P}_{n}$ denote the permutations characterized
by avoiding a set of 1-line patterns $P$.  The most important pattern classes
for this work are the maximally clustered permutations and the freely braided
permutations, characterized by avoiding the patterns from \eqref{e:mc.patterns}
and \eqref{e:fb.patterns}, respectively.  Given a finite set $H$ of
permutations, let $S^{P}(H)$ be the subset of $S^{P}$ consisting of those
permutations that heap-avoid the patterns in $H$.

In \cite[Section 4]{j2} we described how to find a set of 1-line patterns $Q$
such that $S^{P}(\{ h \}) = S^{Q}$, when possible.  Example 11.1 of \cite{b-j1}
shows that some heap patterns $h$ have no such set $Q$.

Let $r(h)$ denote the rank of the symmetric group containing $h$.  Define
$U^{P}(h)$ to be the set of all elements in $S^{P}_{r(h)}$ that heap-contain
$h$.  This is a finite set because the rank is fixed.

\begin{definition}{\bf \cite[Definition 4.2]{j2}}\label{d:ideal}
Let $p \in S^{P}$.  Then, we say that $p$ is an \em ideal pattern in $S^{P}$ \em
if for every $q \in S^{P}_{r(p)+1}$ containing $p$ as a 1-line pattern, we have
that $q$ heap-contains $p$.
\end{definition}

For example, \cite[Theorem 3.8]{t2} implies that $p$ is an ideal pattern in
$S^{\emptyset}$ if $p$ avoids $[2143]$.  Definition~\ref{d:ideal} describes a
finite test that extends to permutations of all ranks according to the following
result.

\begin{theorem}{\bf \cite[Theorem 4.4]{j2}}\label{t:translate_olh}
Suppose $S^{P}(H)$ is the subset of permutations characterized by avoiding a
finite set $P$ of 1-line patterns and heap-avoiding a finite set $H$ of
permutations.  If each of the elements in $P' = \bigcup_{h \in H} U^{P}(h)$ is
an ideal pattern, then $S^{P}(H) = S^{P \bigcup P'}$, so is characterized by
avoiding the permutations in $P \bigcup P'$ as 1-line patterns.
\end{theorem}

We can apply Theorem~\ref{t:translate_olh} to study certain classical
permutation pattern classes using heap-avoidance.  For example, it is
straightforward to verify that
\[ S^{\{[321]\}}(\{ s_5 s_6 s_7 s_3 s_4 s_5 s_6 s_2 s_3 s_4 s_5 s_1 s_2 s_3 \}) =
S^{\{[321], [46718235], [46781235], [56718234], [56781234]\}} \]
using Theorem~\ref{t:translate_olh}.  Similar statements \cite[Corollary
4.5]{j2} hold for the freely braided and maximally clustered permutations.
The permutations that heap-avoid
\[ [46718235] = s_5 s_6 s_7 s_3 s_4 s_5 s_6 s_2 s_3 s_4 s_5 s_1 s_2 s_3 \]
are called \em hexagon-avoiding \em after \cite{b-w}.

Let $H$ be a finite set of connected fully commutative permutations each of
whose heaps contains at least two entries in each internal column in the sense
of Definition~\ref{d:supports} or let $H = \emptyset$.  Suppose $F_n$ is the
subset of the fully commutative permutations on $n$ generators that are
characterized by heap-avoiding the set of patterns from $H$, so $F_n =
S_{n+1}^{[321]}(H) \subset S_{n+1}^{[321]}$.  We define $|F_0|$ to be 1,
corresponding to the empty heap.  Our main result in this section is that we
can transform the generating function $F(x) = \sum_{n \geq 0} |F_n| \ x^n$ to
obtain generating functions for the corresponding freely braided and maximally
clustered pattern classes.  Moreover, the transformation is a rational function
of $F(x)$, so it preserves this important property of the generating function.

We first consider the number of permutations in $F_n$ that have one or both of
the extremal generators $\{ s_1, s_n \}$ present in the heap.  Recall that every
minimal pair of entries in a fully commutative heap must have a distinct
resolution, so in particular any fully commutative heap has at most a single
entry in its leftmost and rightmost columns.  Let $L_n$ be the permutations from
$F_n$ that have $s_n$ present in their heap, and $R_n$ be the set of
permutations from $F_n$ having $s_1$ present in the heap.  By the bijection that
reverses the ordering of the subscripts of the generators, we find that $|L_n| =
|R_n|$.  Let $M_n$ denote the set of permutations in $F_n$ with both $s_1$ and
$s_n$ present in the heap.  Then, we have the following enumerative result.

\begin{lemma}\label{l:pieces}
Let $H$ be a finite set of fully commutative permutations or let $H =
\emptyset$.  Suppose
\[ F(x) = \sum_{n \geq 0} |S_{n+1}^{[321]}(H)| \ x^n. \]
Then,
\[ L(x) = \sum_{n \geq 0} |L_n| \ x^n = F(x) - x F(x) - 1 \]
and
\[ M(x) = \sum_{n \geq 0} |M_n| \ x^n = F(x) - 2 x F(x) + x^2 F(x) - 1. \]
\end{lemma}
\begin{proof}
This follows using inclusion-exclusion together with the observation that $|F_0|
= 1$ and $|F_1| = 2$ so $|L_0| = |M_0| = |M_1| = 0$, which is required by
definition.
\end{proof}

We are now in a position to prove our main enumerative theorem.

\begin{theorem}\label{t:enumerate}
Let $H$ be a finite set of connected fully commutative permutations each of
whose heaps contains at least two entries in each internal column or let $H =
\emptyset$.  Suppose
\[ F(x) = \sum_{n \geq 0} |S_{n+1}^{[321]}(H)| \ x^n, \]
\[ L(x) = F(x) - x F(x) - 1, \text{ and } M(x) = F(x) - 2 x F(x) + x^2 F(x) - 1. \]
Then, we have
\[ \sum_{n \geq 0} |S_{n+1}^{\{[3421], [4231], [4312], [4321]\}}(H)| \ x^n =
F(x) + {{L(x)^2} \over {1 - M(x)}} \]
and
\[ \sum_{n \geq 0} |S_{n+1}^{\{[3421], [4312], [4321]\}}(H)| \ x^n = F(x) + {
{L(x)^2} \over {1 - x - M(x)}}. \]
\end{theorem}
\begin{proof}
Suppose $w$ is an element of $S_{n+1}^{\{[3421], [4312], [4321]\}}(H)$.  By
Proposition~\ref{p:canonical_form}, we may choose a commutativity class $\mathfrak{C}$ of
$w$ so that the
heap of $\mathfrak{C}$ has a specific form where each column either supports a braid
cluster, in which case there are no other generators in that column, or else
the column is not used in any braid cluster.

Let $k$ be the number of braid clusters in $w$.  We partition the columns $\{ 1,
\dots, n \}$ of the heap of $\mathfrak{C}$ into intervals $C_0, B_1, C_1, B_2, \dots, B_k,
C_k$ based on the location of the braid clusters in $w$.  We define the $r$th
interval $C_r$ to be $[p,q]$ where $p$ is the rightmost column of the $r$th
braid cluster and $q$ is the leftmost column of the $(r+1)$st braid cluster,
setting $p=1$ for $r=0$ and $q=n$ for $r=k$.  The $B_i$ then consist of the
internal columns of the $i$th braid cluster.  In particular, if $w$ is
freely braided then every $B_i = \emptyset$.  If $w$ is fully commutative, then
$C_0 = \{ 1, \dots, n \}$.

For example, suppose $w$ is given by the contracted reduced expression
\[ (s_5) (s_1 s_2 s_3 s_4 s_3 s_2 s_1) (s_6 s_5 s_9) (s_7
s_8 s_7) (s_6).\]
Then the heap of $w$ is drawn below, together with its column partition.  The
braid clusters are shown in grey.

{\scriptsize
\[ \xymatrix @=-4pt @! {
& {\hs} & & {\hs} & & {\hf} & & {\hs} & \\
{\hb} & & {\hs} & & {\hs} & & {\hb} \\
& {\hb} & & {\hs} & & {\hs} & & {\hb} & \\
{\hs} & & {\hb} & & {\hf} & & {\hb} & & {\hf} \\
& {\hs} & & {\hb} & & {\hf} & & {\hs} & \\
{\hs} & & {\hb} & & {\hf} & & {\hs} & \\
& {\hb} & & {\hs} & & {\hs} & & {\hs} \\
{\hb} & & {\hs} & & {\hs} & & {\hs} & \\
C_0 \text{]} & \text{[} B_1 & \text{]} & \text{[} C_1 & & &
\text{]} & \text{[} C_2 \\
} \] }

Observe that the entries appearing in columns $B_i$ are completely determined by
the positions of the entries from the rightmost column in $C_{i-1}$ and the
leftmost column in $C_{i}$, as these are the ends of the braid cluster supported
by $B_i$.

Next, we define a map $\pi : \{C_0, C_1, \dots, C_k \} \rightarrow S_{n+1}$ to
project each interval of columns to a permutation.  If $k = 0$, then $\pi(C_0) =
C_0$.  Otherwise, observe that the leftmost column in each $C_i$ for $i > 0$ is
the rightmost column of some braid cluster, so it consists of a single entry.
The rightmost column $q$ in each $C_i$ for $i < k$ is the leftmost column of
some braid cluster, so it consists of two entries, and there can be no entries
between them in column $q-1$ by Definition~\ref{d:mc}.  Hence, we may define a
permutation $\pi(C_i)$, whose heap is obtained from the heap of $w$ restricted to
the columns of $C_i$, by collapsing the two entries in column $q$ to a single
entry.

In the example above, $\pi(C_1)$ has the heap
{\scriptsize \[ \xymatrix @=-4pt @! {
{\hs} & & {\hf} & \\
& {\hf} & & {\hb} \\
{\hb} & & {\hf} & \\
& {\hf} & & {\hs} \\
\text{[} C_1 & & & \text{]} \\
} \] }

Next, we claim that for each $i$,
\begin{equation}\label{e:pieces}
\pi(C_i) \in
\begin{cases}
F_{|C_0|} & \text{ if $k = 0$, } \\
L_{|C_0|} & \text{ if $k > 0$ and $i = 0$, } \\
M_{|C_i|} & \text{ if $k > 0$ and $1 \leq i < k$, } \\
R_{|C_k|} & \text{ if $k > 0$ and $i = k$. }
\end{cases}
\end{equation}
In particular, we show that each $\pi(C_i)$ corresponds to a fully commutative
permutation that heap-avoids the patterns from $H$.

The claim is clear if $k = 0$, so suppose $k > 0$, and let $C_i = [p,q]$ with
$0 < i < k$.  Then $\pi(C_i)$ is fully commutative because every minimal pair
of entries in $\pi(C_i)$ has a distinct resolution by property (2) in
Definition~\ref{d:bc_column_decomposition} from
Proposition~\ref{p:canonical_form}.

Next, suppose there exists an instance of a heap-pattern $h \in H$ among the
entries of $\pi(C_i)$.  Then we must have such a heap-instance in $w$ which is a
contradiction.  This follows because the heap of $w$ with respect to the
commutativity class where the $(i+1)$st braid cluster is of the form
\[ s_{m+k+1} s_{m+k} \dots s_{m+1} s_{m} s_{m+1} \dots s_{m+k} s_{m+k+1} \]
has precisely the entries of $\pi(C_i)$ in columns $C_i$, and so $w$
heap-contains $h$.  As this cannot occur, each $\pi(C_i)$ heap-avoids the
patterns from $H$.

It follows from the construction that if $k > 0$ then $\pi(C_0)$ has an entry in
its rightmost column, $\pi(C_k)$ has an entry in its leftmost column, and every
other $\pi(C_i)$ has entries in both the leftmost and rightmost columns.
Otherwise, $k = 0$, and $\pi(C_0) = C_0 \in F_{|C_0|}$.  Thus, we have proved
(\ref{e:pieces}).

\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\xymatrix @=-4pt @! {
{\hb} & & {\hs} & & \\
& {\hb} & & {\hs} & & \\
{\hs} & & {\hb} & & \\
& {\hs} & & {\hb} & & \\
{\hs} & & {\hs} & & \\
& {\hs} & & {\hb} & & \\
{\hs} & & {\hb} & & \\
& {\hb} & & {\hs} & & \\
{\hb} & & {\hs} & & \\
} \\
\end{tabular}
\end{center}
\caption{Internal columns of a braid cluster}
\label{f:internal_heap}
\end{figure}

Conversely, suppose there exists a fixed a column partition $\{ C_0, B_1, C_1, \ldots, B_k,
C_k \}$ as in the beginning of the proof and we are given either a single heap
$c^{(0)} \in F_{|C_0|}$ if $k=0$ or a sequence of heaps
\begin{equation}\label{e:sequences}
c^{(0)}, b^{(1)}, c^{(1)}, b^{(2)}, \dots, b^{(k)}, c^{(k)}
\end{equation}
where $k > 0$, $c^{(0)} \in L_{|C_0|}$, $c^{(k)} \in R_{|C_k|}$, every other
$c^{(i)} \in M_{|C_i|}$ and each $b^{(i)}$ is a heap fragment on $|B_i|$ columns
with the canonical form shown in Figure~\ref{f:internal_heap}.  Then, we may
form a maximally clustered permutation $w$ with $k$ braid clusters as follows.  If
$k = 0$, take $w$ to be the permutation whose heap is $c^{(0)}$.  Otherwise, for
each $i < k$ we expand the single entry in the rightmost column of $c^{(i)}$ to
a pair of entries.  This has the effect of reversing the $\pi$ map on each
$c^{(i)}$, which we denote by $\pi^{-1}(c^{(i)})$.  Then, glue the columns
together in order to form a single heap, so that the rightmost pair of entries
in each $\pi^{-1}(c^{(i)})$ surrounds the leftmost pair of entries in
$b^{(i+1)}$ if any, or the unique leftmost entry in $\pi^{-1}(c^{(i+1)})$ if
$b^{(i+1)}$ is empty.  Also, each $b^{(i)}$ is glued to $\pi^{-1}(c^{(i)})$ so
that the pair of entries in the rightmost column of $b^{(i)}$ surrounds the
unique entry in the leftmost column of $\pi^{-1}(c^{(i)})$.  We call the
permutation to which this heap corresponds $w$.

Every minimal pair of entries $x$ and $y$ from an internal column of
$\pi^{-1}(c^{(i)})$ must have a distinct resolution because $c^{(i)}$ is
fully commutative.  Hence, we observe that the heap we constructed is reduced
and maximally clustered because it has the form given in
Proposition~\ref{p:canonical_form}.

Next, observe that if $w$ heap-contains a pattern $h \in H$ then the pattern
instance must lie in one of the $c^{(i)}$ To see this, let $x$ and $y$ be a
minimal pair of entries lying in column $i \in [2, n-1]$ of $h$.  Because $h$
is fully commutative, $x$ and $y$ have a distinct resolution, so there exist
entries from columns $i+1$ and $i-1$ lying strictly between $x$ and $y$ in the
heap poset.  However, no column in the heap of any commutativity class for a
braid cluster has this property by the uniqueness statement in
Definition~\ref{d:mc}, since each commutativity class for a braid cluster has a
reduced expression representative that is itself a braid cluster.  Thus, no
heap-instance of $h$ uses the internal columns of a braid cluster.  Since we
are assuming that each $c^{(i)}$ heap-avoids the patterns from $H$, we have
that $w$ heap-avoids the patterns from $H$.

Finally, note that if we compose the constructions we have given above in either
order, then we obtain the identity.  Hence, we have shown a bijection and the
enumerative formulas follow.  $F(x)$ contributes terms of the form where $k =
0$.  Using Lemma~\ref{l:pieces}, we obtain the generating function that counts
sequences of the form given in (\ref{e:sequences}) as the product
\[ L(x) \cdot {1 \over {1-x}} \cdot {1 \over {1 - M(x) \cdot {1 \over {1-x}}}}
\cdot L(x) \]
because each term of the sequence in (\ref{e:sequences}) is independent.
Multiplying the numerator and denominator of the third factor by $(1-x)$, we
obtain the second result.  In the case where $w$ is freely braided, each
$b^{(i)}$ term is $\emptyset$, so the corresponding generating function is
simply
\[ L(x) \cdot {1 \over {1 - M(x)}} \cdot L(x) \]
which gives the first result.
\end{proof}

\begin{corollary}\label{c:enumerate}
Let $H$ be a finite set of connected fully commutative permutations each of
whose heaps contains at least two entries in each internal column.
If $F(x) = \sum_{n \geq 0} |S_{n+1}^{[321]}(H)| \ x^n$
is a rational (respectively, algebraic) generating function, then
\[ \sum_{n \geq 0} |S_{n+1}^{\{[3421], [4231], [4312], [4321]\}}(H)| \
x^n \text{\ \ and \ \ } \sum_{n \geq 0} |S_{n+1}^{\{[3421], [4312], [4321]\}}(H)| \ x^n. \]
are also rational (respectively, algebraic).
\end{corollary}

Using Theorems~\ref{t:translate_olh} and \ref{t:enumerate}, we can enumerate
several interesting classes characterized by permutation pattern avoidance.
Note that we index the coefficients in the generating functions by the rank of
the Coxeter group rather than by the number of entries appearing in the 1-line
notation.

\bigskip
\noindent
\begin{tabular}{|p{0.95in}|l|l|}
\hline
& Generating function & Initial sequence \\
\hline
$[321]$-avoiding & ${{1 - 2x - \sqrt{1-4x}} \over 2x^2}$ &
{\tiny $1 + 2
x + 5 x^2 + 14 x^3 + 42 x^4 + 132 x^5 + 429 x^6 + 1430 x^7 + \dots$ }  \\
\hline
$L(x)$ \ \ \ \ \ \ for $H = \emptyset$ & ${{1 - 3x - (x-1) \sqrt{1-4x}} \over 2x^2}$ & {\tiny
$x + 3 x^2 + 9 x^3 + 28 x^4 + 90 x^5 + 297 x^6 + 1001 x^7 + \dots$ } \\
\hline
$M(x)$ \ \ \ for $H = \emptyset$ & ${{1 - 4x + 3x^2 -2x^3 - (x-1)^2 \sqrt{1-4x}} \over 2x^2}$
& {\tiny $2 x^2 + 6 x^3 + 19 x^4 + 62 x^5 + 207 x^6 + 704 x^7 + \dots$ } \\
\hline
Freely-braided & ${{2x-2x^2-2x \sqrt{1-4x}} \over {-1+4x-x^2+2x^3+(x-1)^2
\sqrt{1-4x}}} $ & {\tiny $1 + 2x + 6x^2 + 20x^3 + 71 x^4 + 260 x^5 +
971x^6 + 3674x^7 + \dots$ } \\
\hline
Maximally-clustered & ${{2x} \over {-1 + 4x - 2x^2 + \sqrt{1-4x}}}$ &
{\tiny $1+2x+6x^2+21x^3+78x^4+298x^5+1157x^6+4539x^7+\dots$ } \\
\hline
$[321]$-hexagon avoiding & ${{-x^5+x^4+3x^3-4x^2+4x-1} \over
{x^6-4x^5-4x^4+9x^3-11x^2+6x-1}}$ & {\tiny $1 + 2 x + 5 x^2 + 14 x^3 + 42
x^4 + 132 x^5 + 429 x^6 + 1426 x^7 + \dots$ } \\
\hline
$L(x)$ for $H = \{[46718235]\}$ & ${{2x^5+2x^4-2x^3+3x^2-x} \over
{x^6-4x^5-4x^4+9x^3-11x^2+6x-1}}$ & \tiny { $x + 3 x^2 + 9 x^3 + 28 x^4 +
90 x^5 + 297 x^6 + 997 x^7 + \dots$ } \\
\hline
$M(x)$ for $H = \{[46718235]\}$ & ${{-x^7+2x^6+4x^5-5x^4+6x^3-2x^2} \over
{x^6-4x^5-4x^4+9x^3-11x^2+6x-1}}$ & {\tiny $2 x^2 + 6 x^3 + 19 x^4 + 62
x^5 + 207 x^6 + 700x^7 + \dots$ } \\
\hline
Freely-braided hexagon-avoiding & ${{-x^6-2x^5+2x^4+x^3-3x^2+4x-1} \over
{x^7-x^6-8x^5+x^4+3x^3-9x^2+6x-1}}$ & {\tiny $1 + 2x + 6x^2 + 20x^3 + 71
x^4 + 260 x^5 + 971x^6 + 3670x^7 + \dots$ } \\
\hline
Maximally-clustered hexagon-avoiding & ${{3x^5+x^4-5x^3+7x^2-5x+1} \over
{-3x^6+4x^5+8x^4-14x^3+15x^2-7x+1}}$ & {\tiny
$1+2x+6x^2+21x^3+78x^4+298x^5+1157x^6+4535x^7+\dots$ } \\
\hline
\end{tabular}
\bigskip

\begin{remark}
The first line is the Catalan generating function which appears in this context
by \cite{simion-schmidt}, while the sixth line is Theorem~\ref{t:s-w} due to
\cite{s-w}.  The freely braided permutations have previously been enumerated in
\cite{mansour}, while the other formulas seem to be new.  They follow from
Theorem~\ref{t:translate_olh} together with Lemma~\ref{l:pieces} and
Theorem~\ref{t:enumerate} by taking $H = \emptyset$ and $H = \{ [46718235] \}$,
respectively.
\end{remark}

\begin{corollary}
The number $b_n$ of freely-braided hexagon-avoiding permutations in $S_n$ satisfies the
recurrence
\[ b_{n+1} = 6 b_{n} - 9b_{n-1} + 3b_{n-2} + b_{n-3} - 8b_{n-4} - b_{n-5} +
b_{n-6} \]
and the number $m_n$ of maximally-clustered hexagon-avoiding permutations in $S_n$ satisfies the
recurrence
\[ m_{n+1} = 7 m_{n} - 15m_{n-1} + 14m_{n-2} - 8m_{n-3} - 4m_{n-4} + 3m_{n-5} \]
for all $n \geq 9$, with initial conditions given by Figure~\ref{f:mc.enum}.
\end{corollary}

\bigskip
\section{Diamond reductions}\label{s:diamond}

In this section, we restrict to considering fully commutative permutations.

\begin{definition}
Suppose that $h$ is a connected, fully commutative permutation whose heap
contains at least two entries in each internal column.  We say that every
minimal pair of entries together with the entries of their distinct resolution
forms a \em minimal diamond \em inside the heap of $h$.  Form a new heap whose
entries correspond to minimal diamonds.  The poset structure of the new heap is
inherited from the poset structure on the minimal diamonds in the heap of $h$ by
taking the transitive closure of the relation that two minimal diamonds are
related if they share an edge.  The heap obtained in this way from $h$ is called
the \em diamond reduction \em of $h$.
\end{definition}

Pictorially, the heap of the diamond reduction of $h$ is obtained from the heap
of $h$ by adding entries at the centers of all minimal diamonds and then erasing
the heap of $h$.

\begin{example}\label{ex:3hex}
The diamond reduction of the hexagon $s_5 s_6 s_7 s_3 s_4 s_5 s_6 s_2 s_3 s_4
s_5 s_1 s_2 s_3$ is the \em 3-hexagon \em $s_4 s_5 s_2 s_3 s_4 s_1 s_2$.
\[
\xymatrix @=3pt @! {
 & & \hf \ar@{-}[dl] \ar@{-}[dr] & & \hf \ar@{-}[dl] \ar@{-}[dr] \\
 & \hf \ar@{-}[dr] \ar@{-}[dl] & \hb & \hf \ar@{-}[dl] \ar@{-}[dr] & \hb & \hf \ar@{-}[dl] \ar@{-}[dr]\\
 \hf \ar@{-}[dr] & \hb & \hf \ar@{-}[dr] \ar@{-}[dl] & \hb & \hf \ar@{-}[dl] \ar@{-}[dr] & \hb & \hf \ar@{-}[dl] \\
 & \hf \ar@{-}[dr] & \hb & \hf \ar@{-}[dl] \ar@{-}[dr] & \hb & \hf \ar@{-}[dl] \\
 & & \hf & & \hf \\
}
\parbox[t]{1in}{ \vspace{.3in}
\hspace{.3in} $\longrightarrow$ }
\xymatrix @=3pt @! {
 & & & \\
 & \hb & & \hb \\
 \hb & & \hb & & \hb \\
 & \hb & & \hb \\
}
\]
\end{example}

\begin{proposition}\label{p:dr_bij}
The diamond reduction is a bijection from the set of connected fully commutative
heaps on $n$ columns with at least two entries in each internal column to the
set of connected fully commutative heaps on $n-2$ columns.
\end{proposition}
\begin{proof}
Let $h$ be a connected fully commutative heap on $n$ columns with at least two
entries in each internal column.  The diamond reduction $g$ of $h$ gives a
reduced heap because we can represent the minimal diamonds by their maximal
entries (rather than the centers) and the result is a convex subposet of the
heap of $h$.  The heap of $g$ is also connected since every internal column of
the heap of $h$ has a minimal diamond.  The diamond reduction can be reversed by
forming a heap with minimal diamonds centered at the heap entries of $g$.
Hence, the diamond reduction of $h$ is fully commutative because any short-braid
instance in the heap of $g$ would imply the existence of a short-braid in $h$
when we consider the minimal diamonds centered at the entries of the heap of
$g$.  We also reduce the number of columns by 2 since only the internal columns
of the heap of $h$ support minimal diamonds that become entries in the heap of
$g$.
\end{proof}

Our main goal in this section is to describe how the generating functions for
$\{|S_{n+1}^{[321]}(h)|\}$ and $\{|S_{n+1}^{[321]}(g)|\}$ are related when $g$
is the diamond reduction of $h$.

\begin{lemma}\label{l:dr_connected}
Let $H$ be a set of connected, fully commutative permutations.
Suppose
\[ F(x) = \sum_{n \geq 0} |S_{n+1}^{[321]}(H)| \ x^n \text{\ \  and \ \ }
F_{c}(x) = \sum_{n \geq 0} |\{ p \in S_{n+1}^{[321]}(H) : \text{ $p$ is connected } \}| \ x^n, \]
with $F(0) = 1$ and $F_{c}(0) = 0$.
Then the generating functions are rationally related according to
\[ F(x) = {{1+F_{c}(x)} \over {1-x-x F_{c}(x)}} \text{\ \  and \ \ }
F_{c}(x) = { {F(x)-x F(x)-1} \over {1+x F(x)} }. \]
\end{lemma}
\begin{proof}
To see this, observe that every not-necessarily-connected $H$-avoiding heap on
$n$ columns decomposes uniquely into connected components, each of which is
counted by $F_{c}(x)$.  Conversely, if we have an ordered collection
$\mathcal{C}$ of connected permutations that all heap-avoid the patterns from
$H$, then we can place them together from left to right, to obtain a heap which
has the heaps from $\mathcal{C}$ as its connected components.

The generating function identity that we obtain from this observation is
\[ F(x) = {1 \over {1-x}} + {1 \over {1-x}} \cdot F_{c}(x) \cdot {1 \over {1 - {
{x \over {1-x}} F_{c}(x) }}} {1 \over {1-x}}. \]
Specifically, we either have some connected components, or none.  The initial
${1 \over {1-x}}$ term corresponds to the case where we have none.  Otherwise,
we have zero or more initial empty columns, followed by at least one connected
component, followed by zero or more copies of empty columns together with other
connected components, finally followed by zero or more empty columns on the
right side of the heap.  This is the second term.  This formula simplifies to
the one given in the statement and can be inverted.
\end{proof}

One fundamental subclass of the fully commutative permutations are those with no
minimal diamonds at all.  It is straightforward to see that this corresponds to
having at most one entry in each column, or equivalently to heap-avoiding $s_2
s_1 s_3 s_2$.  This pattern class was previously enumerated in \cite{fan1} and
\cite{west2}.  Also, \cite{t3} enumerates these permutations using the statistic of
Coxeter length.

\begin{lemma}\label{l:diamond_avoiding}
We have
\[ \sum_{n \geq 0} |S_{n+1}^{[321], [3412]}| \ x^n = \sum_{n \geq 0} |S_{n+1}^{[321]}(s_2 s_1 s_3 s_2)| \ x^n = { {1-x} \over {1 - 3x + x^2 }}. \]
\end{lemma}
\begin{proof}
The second equality follows from Theorem~\ref{t:translate_olh}.  Let $w$ be a
connected element of $S_{n+1}^{[321]}(s_2 s_1 s_3 s_2)$.  Then the heap of $w$
is a lattice path consisting of $n-1$ steps that are either ``up'' or ``down.''
As each step is independent, the generating function for these is ${{x} \over
{1-2x}}$.  Applying Lemma~\ref{l:dr_connected} gives the result.
\end{proof}

We are now in a position to state and prove our main result in this section.

\begin{theorem}\label{t:main_diamond}
Suppose that $h$ is a connected, fully commutative permutation whose heap
contains at least two entries in each internal column and let $g$ be the diamond
reduction of $h$.  If
\[ G(x) = \sum_{n \geq 0} |S_{n+1}^{[321]}(g)| \ x^n \text{, \ \  \ \ }
G_{c}(x) = \sum_{n \geq 0} |\{ p \in S_{n+1}^{[321]}(g) : \text{ $p$ is
connected } \}| \ x^n, \]
and $F(x) = \sum_{n \geq 0} |S_{n+1}^{[321]}(h)| \ x^n$,
with $G(0) = 1 = F(0)$ and $G_{c}(0) = 0$,
then the generating functions are rationally related according to
\begin{equation}\label{e:dr_main}
F(x) = { {1-x-x G_{c}(x)} \over {1-3x+x^2+(x^2-x)G_{c}(x)}}
= { {1} \over {1-2x-x^2 G(x)}}.
\end{equation}
\end{theorem}
\begin{proof}
Suppose $w \in S_{n+1}^{[321]}(h)$.  We assign each of the columns $\{1, \ldots n\}$
of the heap of $w$ to intervals $E_0, D_1, E_1, \ldots, D_k, E_k$ according to
whether the column supports more than one entry.  The intervals $D_i$ are
defined to be precisely those that support maximal connected fully commutative
subheaps of the heap of $w$ such that the internal columns of $D_i$ each support
at least two entries.  The remaining intervals $E_i$ consist of columns that
support at most one entry.  In particular, the $E_i$ may include empty columns.

For example, suppose
\[ w = s_1 s_2 s_{12} s_8 s_9 s_{10} s_{11} s_5 s_6 s_7 s_8 s_4 s_5 s_6. \]
Then the heap of $w$ is drawn below, together with its column assignments.
{\scriptsize
\[ \xymatrix @=-4pt @! {
& & \\
& {\hs} & & {\hs} & & {\hs} & & {\hs} & & {\hs} & & {\hf} \\
{\hs} & & {\hs} & & {\hs} & & {\hf} & & {\hf} & & {\hf} & & {\hf} \\
& {\hs} & & {\hs} & & {\hf} & & {\hf} & & {\hf} & \\
{\hs} & & {\hs} & & {\hf} & & {\hf} & & {\hf} & \\
& {\hs} & {\hf} & {\hs} & & {\hf} & & {\hs} & & {\hs} \\
{\hs} & {\hf} & {\hs} & & {\hs} & & {\hs} & & {\hs} & \\
& \text{[} & E_0 & & \text{]} &     & &\text{[$E_1$]}   &     & \text{[} & E_2 & & \text{]} \\
&             &  & & \text{[} & D_1 & &\text{][} & D_2 & \text{]} &  & & \\
} \] }

Let $w|_{D_i}$ denote the restriction of the heap of $w$ to the columns in the
interval $D_i$.  By the hypotheses given, each $w|_{D_i}$ must heap-avoid $h$.
Moreover, we can form the diamond reduction of $w|_{D_i}$ whose heap must be
connected, heap-avoids $g$, and has 2 fewer columns than $w|_{D_i}$.  Each
$w|_{E_i}$ must heap-avoid $s_2 s_1 s_3 s_2$ and contains entries in the
extremal columns of $E_i$ that are shared with $D_{i}$ or $D_{i+1}$.

Conversely, suppose there exists a fixed column assignment $E_0, D_1, E_1,
\ldots, D_k, E_k$  and we are given a sequence of heaps $e_0, d_1, e_1, \ldots,
d_k, e_k$ such that $e_0 \in S_{|E_0|+1}^{[321]}(s_2 s_1 s_3 s_2)$ with one
entry in the rightmost column of $E_0$, $d_i \in S_{|D_i|-1}^{[321]}(g)$ and
$d_i$ is connected for $1 \leq i \leq k$, $e_i \in S_{|E_i|+1}^{[321]}(s_2 s_1
s_3 s_2)$ for $1 \leq i \leq k-1$ with one entry in each of the extremal columns
of $E_i$ and $e_k \in S_{|E_i|+1}^{[321]}(s_2 s_1 s_3 s_2)$ with one entry in
the leftmost column of $E_k$.  Then we can apply Proposition~\ref{p:dr_bij} to
form the reverse diamond reduction $\widetilde{d_i}$ of $d_i$ and glue these
heaps together to obtain an element of $S_{n+1}^{[321]}(h)$.  Specifically, we
identify the unique entry in the rightmost column of $e_i$ with the unique entry
in the leftmost column of $\widetilde{d_{i+1}}$ for each $0 \leq i \leq k-1$,
and we also identify the unique entry in the leftmost column of $e_i$ with the
unique entry in the rightmost column of $\widetilde{d_{i}}$ for each $1 \leq i
\leq k$.

If we compose the constructions we have given in either order, we obtain the
identity.  Hence, we have shown a bijection.

Let $E(x)$ be the generating function appearing in
Lemma~\ref{l:diamond_avoiding}.  Define $E_{LR}(x) = E(x) - x E(x) - 1$ and
$E_{M}(x) = E(x) - 2x E(x) + x^2 E(x) - 1 + x$.  It is straightforward to verify
using inclusion-exclusion that $E_{LR}(x)$ counts the number of $s_2 s_1 s_3
s_2$-avoiding heaps with one entry in at least one of the two extremal columns,
and $E_{M}(x)$ counts the number of $s_2 s_1 s_3 s_2$-avoiding heaps with one
entry in both of the extremal columns.

Putting all of these observations together, we have the generating function
identity
\begin{equation}\label{e:main_diamond}
F(x) = E(x) + E_{LR}(x) {1 \over {1 - G_{c}(x) E_{M}(x)}} G_{c}(x) E_{LR}(x).
\end{equation}
Here, $E(x)$ contributes terms of the form where $k = 0$.  This formula
simplifies to the first equality given in the statement.  The second equality
follows from Lemma~\ref{l:dr_connected}.
\end{proof}

\begin{remark}
In the nicest cases, this result allows us to reduce the problem of enumerating
permutations that avoid a given heap pattern to counting lattice paths that
avoid a certain consecutive subpath.  This problem is well known to have a
rational generating function and the transfer matrix method can be used to find
the generating function explicitly.  See \cite[Example 4.7.5]{ec1} for details.
\end{remark}

\begin{example}
\[ \sum_{n \geq 0} |S_{n+1}^{[321], [356124], [456123]}| \ x^n = \sum_{n \geq 0} |S_{n+1}^{[321]}(s_4 s_5 s_2 s_3 s_4 s_1 s_2 s_3)| \ x^n = { {1-3x+2x^2-x^3} \over {1 - 5x + 7x^2 -4x^3+x^4}}. \]

The first equality follows from Theorem~\ref{t:translate_olh}.
The heap pattern $p = s_4 s_5 s_2 s_3 s_4 s_1 s_2 s_3$ has a diamond reduction to $q = s_1 s_3 s_2$.
\[
\xymatrix @=3pt @! {
 & & \hf \ar@{-}[dl] \ar@{-}[dr] & & \\
 & \hf \ar@{-}[dl] \ar@{-}[dr] & \hb & \hf \ar@{-}[dl] \ar@{-}[dr] \\
 \hf \ar@{-}[dr] & \hb & \hf \ar@{-}[dl] \ar@{-}[dr] & \hb & \hf \ar@{-}[dl] \\
 & \hf & & \hf \\
}
\parbox[t]{1in}{ \vspace{.3in}
\hspace{.3in} $\longrightarrow$ }
\xymatrix @=3pt @! {
 & & & \\
 \hs & & \hb & & \hs \\
 & \hb & & \hb \\
}
\]
The number of connected permutations that heap-avoid $q$ is the same as the
number of lattice paths that never contain a consecutive up-down subpath.  It is
straightforward to see that there are $n$ such lattice paths that use $n$ nodes.
The corresponding generating function is $G_{c}(x) = {x \over {(1-x)^2}}$ and
substituting this into Equation~(\ref{e:dr_main}) yields the result.
\end{example}

\begin{example}
\[ \sum_{n \geq 0} |S_{n+1}^{[321], [46718235], [46781235], [56718234], [56781234]}| \ x^n
= \sum_{n \geq 0} |S_{n+1}^{[321]}(s_5 s_6 s_7 s_3 s_4 s_5 s_6 s_2 s_3 s_4 s_5 s_1 s_2 s_3)| \ x^n \]
\[ = { {1-4x+4x^2-3x^3-x^4+x^5} \over {1 - 6x + 11x^2 -9x^3+4x^4+4x^5-x^6}}. \]
recovering Theorem~\ref{t:s-w}

The first equality follows from Theorem~\ref{t:translate_olh}.
The hexagon pattern has a diamond reduction to $q = s_4 s_5 s_2 s_3 s_4 s_1 s_2$
as shown in Example~\ref{ex:3hex}.  However, the diamond reduction of $q$ is
$s_1 s_3$ which is disconnected.  Hence, we must enumerate the permutations
heap-avoiding $q$ directly.  We use the same column assignment as in the proof
of Theorem~\ref{t:main_diamond}.  In order to heap-avoid $q$, the intervals of
columns supporting diamonds must restrict to fully commutative permutations
whose diamond reduction is a connected monotonic lattice path.  Also, we can no
longer glue two diamond-containing regions together along a trivial lattice path
with one column.  Hence, we must modify the $E_{M}(x)$ term of
Equation~(\ref{e:main_diamond}) by subtracting $x$.

The monotonic lattice paths are counted by $G_{c}(x) = {{2x} \over {1-x}} -x$.
Substituting this into Equation~(\ref{e:main_diamond}) along with $E_{M}(x) =
E(x) - 2x E(x) + x^2 E(x) - 1$ gives the generating function for all
$q$-heap-avoiding permutations.  Applying Theorem~\ref{t:main_diamond} then
yields the result.
\end{example}

\bigskip
\section*{Acknowledgments}

We thank Sara Billey, Jozsef Losonczy, and Richard Green for many useful
suggestions, as well as Julian West for introducing us to Olivier Guibert's
code \cite{guibert} for permutation pattern enumeration.

redpajama_set_name: RedPajamaArXiv


text:
Police reports show Corpus Christi woman may have been abused years before fatal shooting
Corpus Christi police reports show a history of abuse between Priscilla Davila and her live-in boyfriend, Larry Moore. Here's what they show.
Police reports show Corpus Christi woman may have been abused years before fatal shooting Corpus Christi police reports show a history of abuse between Priscilla Davila and her live-in boyfriend, Larry Moore. Here's what they show. Check out this story on caller.com: https://www.caller.com/story/news/local/behind-broken-doors/2019/04/12/reports-show-corpus-christi-woman-abused-before-she-killed/3298724002/
Alexandria Rodriguez, Corpus Christi Caller Times Published 3:16 p.m. CT April 12, 2019 | Updated 7:27 p.m. CT April 12, 2019
On the second floor of the Nueces County Courthouse, Rosa Maria Cervantes and Shirley Esparza work to help victims of crimes.
Corpus Christi police officers were called out to the home of Priscilla Davila several times, her brother said. Most of the calls came after she told police she was abused.
The last time police arrived at the 32-year-old's door, she was dead. It was the night of March 7.
Corpus Christi police officers investigate a shooting in the Northside area, Thursday, March 7, 2019. Larry Darnell Moore is facing two counts of suspicion of capital murder and was in Nueces County Jail on Friday. (Photo: Annie Rice/Caller-Times)
Police officers were called to an apartment complex in the 3000 block of Antelope Street for reports of a disturbance possibly involving a knife, Police Chief Mike Markle said that night.
Officers arrived to the complex, where they were met with gunfire. They hunkered down, radioed for help and later took the alleged gunman into custody.
Police entered an apartment moments later, where they discovered the bodies of two women. One of them was Davila. The other was Sonia Quintanilla Trejo, 37.
More: Prosecutor: Man will not face death penalty in Corpus Christi double homicide
"She was a victim of something tragic, an innocent bystander in her own home, something horribly she did not deserve," a GoFundMe Page for Trejo said. "She was shot by trying to help (as always) a friend, (her neighbor) in need of her protection at the time."
A GoFundMe Page was created for Sonia Quintanilla Trejo, 37. Trejo was killed March 7, 2019, at her apartment in the 3000 block of Antelope Street. (Photo: Screenshot)
Davila's live-in boyfriend, 41-year old Larry Moore, was in police custody on suspicion of murder. He faces two counts of suspicion of capital murder and two counts of attempted capital murder.
Police reports show officers spoke to Moore and Davila many times before the fatal shootings.
A history of abuse
Davila's brother, Juan Davila, said his sister was a victim of domestic violence.
"Stop domestic violence or abuse ... there's always a place for the victims to go to," Juan Davila said. "Let my sister's outcome be an eye-opener for everyone who's in an abusive relationship. That they need to get out (before) it's too late."
More: Family: Woman fatally shot on Corpus Christi Northside was victim of domestic violence
Corpus Christi police reports reveal the two had been together on-and-off for about 12 years. They had two children together.
Reports with accusations of abuse date back to more than 10 years ago. A report was taken in June 2007. Davila told police Moore slapped her after she asked for money.
"Apparently the victim asked the suspect for some money in order to obtain a food handler's permit and to watch their daughter while she attended the classes," the report states. "The suspect became upset and began arguing with the victim outside her front porch. The suspect then slapped the victim who refused medical treatment at the scene."
Davila had bruising to her right eye and a swollen upper lip. Police took photos of her injuries.
A GoFundMe Page was created for Priscilla Davilla, 32. (Photo: Screenshot)
The assault case — a Class A misdemeanor — was dismissed in 2009, Nueces County Court records show.
In July 2016, police were called again. Davila told officers she was beaten. She said she didn't know who attacked her, but thought Moore's ex-girlfriend arranged the beating, reports state.
"Davila stated that she was getting out of her vehicle, when three (women) approached her," the report reads. "Davila stated that she has never seen any of the three (women), but two of the three (women) attacked her and began stomping her, kicking her and punching her."
Davila had a bump on the right side of her head and was bleeding between her eyebrows, the report states. She could not give any more information about the women.
She was taken to the hospital by medics where doctors administered testing.
Doctors later told police Davila suffered a collapsed lung and "sustained a hole to her lung when she was attacked," the report reads.
Davila's brother disputes she was attacked by three women. He said Moore hurt Davila in that incident.
"He scared her (into telling) the cops that," Juan Davila said.
But Priscilla Davila never told police she was afraid of her boyfriend, reports show.
Larry Moore (Photo: Contributed photo)
A year later, she did tell police Moore threatened her in April 2017.
In that incident Davila called police after she was punched in the face. She told police Moore was the one who punched her once she was at an area hospital.
More: Double homicide on Corpus Christi's Northside: Here's what we know about women, suspect
Davila told police the two argued and she attempted to hit Moore. Moore then "punched her one time with a closed fist to the left side of her face, causing her injury," the report reads.
A doctor told police the punch caused a fracture in Davila's orbital socket.
Police conducted a lethality assessment and Davila said she would stay with a relative for the night.
She told police "she was not going to press charges because Larry had threatened her and told her that if she made a report that things were going to get worse for her," the report reads.
She also said the incident was the first time an argument between the two turned violent, the report states.
The report never says Davila was afraid of Moore. Police asked Davila if she wanted to speak to someone with the women's shelter but "she declined," the report reads.
Police spoke to Davila again in October and asked about pressing charges against Moore. Davila declined again and the case was closed, the report states.
The shooting
Police were called to Davila's home for the last time March 7, 2019. Officers arrived and found Davila and her neighbor dead.
Moore's attorney, Kenneth Botary, said Davila and Moore got into a physical argument before the shooting.
A Corpus Christi police officer shines a flashlight into a police vehicle on Thursday, March 7, 2019. A man is in custody after two women died of apparent gunshot wounds. (Photo: Annie Rice/Caller-Times)
It is unknown what the couple argued about. At some point, Davila ran to a neighbor's apartment. Moore followed, Botary said.
Another altercation occurred and shots were fired.
Officials with the Nueces County Medical Examiner's Office said Davila and Trejo had gunshot and stab wounds. It is unknown what was used to stab the women. The cause of the women's deaths also is unknown pending an autopsy.
Moore opened fire on the police vehicle, hoping the officers would shoot and kill him, Botary said.
Once outside the apartment and in custody, Moore could be heard yelling "one of you (expletive) needs to put a bullet in my head."
A motive for the fatal shootings of Davila and Trejo is still unknown.
Being abused?
Contact The Purple Door, a nonprofit agency that offers free services to victims and survivors of family violence and sexual assault.
Hotlines: 1-800-580-4878 or 361-881-8888
Alexandria Rodriguez covers breaking news and crime in the Corpus Christi area. Send news tips to alexandria.rodriguez@caller.com and follow her on Twitter @Caller_AR.
Double homicide on Corpus Christi's Northside
Corpus Christi police officers investigate a shooting in the Northside area, Thursday, March 7, 2019. Larry Darnell Moore is facing two counts of suspicion of capital murder and was in Nueces County Jail on Friday. Annie Rice/Caller-Times
Corpus Christi police officers shine their flashlights on evidence markers during an investigation in a shooting in the Northside area, Thursday, March 7, 2019. Police say they were met with gunfire while at a call at a Northside apartment complex. Annie Rice/Caller-Times
Corpus Christi Police Chief Mike Markle speaks to members of the media related to an investigation of a shooting in the Northside area, Thursday, March 7, 2019. Markle said officers were called about a stabbing at an apartment in the 3000 block of Buffalo Street about 7:27 p.m. that ended up being a double homicide. Annie Rice/Caller-Times
A Corpus Christi firefighter, top left, and a Corpus Christi police officer, bottom right, investigate a shooting in the Northside area, Thursday, March 7, 2019. After an encounter with the alleged gunman, officers later discovered the bodies of two women in one of the units. Annie Rice/Caller-Times
Corpus Christi police car lights flash in the Northside area, Thursday, March 7, 2019. A 41-year-old man who is accused of shooting at Corpus Christi police officers confessed to killing his girlfriend during an argument, police say. Annie Rice/Caller-Times
Read or Share this story: https://www.caller.com/story/news/local/behind-broken-doors/2019/04/12/reports-show-corpus-christi-woman-abused-before-she-killed/3298724002/

redpajama_set_name: RedPajamaCommonCrawl


The Public Works Administrative Office is located at 500 South 4th Avenue, in Brighton Colorado.
Street, Fleet, Traffic Engineering and Capital Roadway Design and Construction Inspections are coordinated by the Street Administrative Division of the Department.
Please contact us at 303-655-2034 with questions or concerns regarding Street Administrative Policy, Capital Improvements or Roadway issues.


text:
Here we will give info on our ongoing campaigns. Please refer to our blog, press releases and social media pages for the most up-to-date information.
Support for wider measures for safe walking, rolling and cycling in Inverness – Covid response
The Highland Council have made proposals to respond to the need for social distancing during the Covid-19 pandemic. Their Spaces for People proposals are a good start but we feel they could go further. You can consult the proposals and comment by clicking here. Join us to campaign for a more ambitious vision to get Inverness moving out of cars. Click here to sign the petition.
A82 active travel corridor
We are actively campaigning for an improvement to proposals from Bear Scotland to create an active travel route from the Longman roundabout near the Kessock Bridge to Tomnahurich Bridge. Initial proposals were very poor and we have gained political support for a rethink which could transform this vital through-route across Inverness.
Safe streets around schools
We want to live in a city where our children can get to and from school safely and without fear of over-domination by motor traffic. We will work with schools on a case-by-case basis with involvement from local parents to help change this culture. If you want to run an event or improve the streets around your school, please get in touch!
Accessing Inverness
We support plans to improve Academy Street and the area around the railway station in Inverness to make it safer and more attractive for cyclists and pedestrians, while recognising that there are some issues to be resolved during future design stages.
Maritime Heritage Trail
Highland Cycle Campaign is fully supportive of plans to
transform Shore Street, Cromwell Road and Longman Drive to make a more cycle
and pedestrian friendly route.
Kessock Bridge cycle path
We are looking at potential improvements to the area around the Kessock Bridge with a medium to long-term approach. Signing of diversion routes on the National Cycle Network route is a regular problem, as is regular maintenance of the approach path. We have already had an information/map panel installed with help from Sustrans ArtRoots fund and had discussions regarding improvements to the landscaping and approach once the Longman roundabout is redeveloped.


text:
Q: Dynamically render following weekdays depending on what the current 'day' is I am creating a data visualisation and am trying to get the days 2-7 to render dynamically depending on the current day.
The dynamic function to get the current day is working and is saved as the variable called 'day'. However, I don't know what logic to use to create the following weekdays in my google chart depending on what the 'day' variable is.
// Temporary array of placeholder dates for the last 9 days

 let tempArray = [{time: "2020-11-28T04:13:30.000Z", value: 30},
                    {time: "2020-11-27T04:13:40.000Z", value: 77},
                    {time: "2020-11-26T04:13:50.000Z", value: 96},
                    {time: "2020-11-25T04:14:00.000Z", value: 64},
                    {time: "2020-11-24T04:14:10.000Z", value: 58},
                    {time: "2020-11-23T04:14:20.000Z", value: 45},
                    {time: "2020-11-22T04:14:30.000Z", value: 24},
                    {time: "2020-11-21T04:14:40.000Z", value: 19},
                    {time: "2020-11-20T04:14:50.000Z", value: 44},
                    {time: "2020-11-19T04:15:00.000Z", value: 49}]

// Function to turn ISO date format into an integer from 0-6 to get the day

    function getDayWeek(timeData, i){
      let date = timeData[i].time
      switch (new Date(date).getDay()) {
        case 0:
          day = "Sun";
          break;
        case 1:
          day = "Mon";
          break;
        case 2:
          day = "Tue";
          break;
        case 3:
          day = "Wed";
          break;
        case 4:
          day = "Thu";
          break;
        case 5:
          day = "Fri";
          break;
        case 6:
          day = "Sat";
      }
    }

...

// Render Google Chart

function drawChart(dayData) {

...

// Fires function

      getDayWeek(tempArray, 0)

// Graph row render

// 'day' variable returns Sat
// Code is missing the logic/function for dynamically rendering the following days

      let color = '#2196F3'
      var data = google.visualization.arrayToDataTable([
        ["Day", "Battery", { role: "style" } ],
        [day, data1, '#B33D3C'],
        [day2, data2, color],
        [day3, data3, color],
        [day4, data4, color],
        [day5, data5, color],
        [day6, data6, color],
        [day7, data7, color],
      ]);
...

  }

So, how do I dynamically render following weekdays depending on what the current 'day' is?
Let me know if you need me to explain further.

A: As I told you in comment, using momentJS is really nice. This one or another date library but not the native JS Date
Some example to get started
const now = moment();
const now_as_date = now.toDate(); // give a valid JS Date

const dayName = now.format('dddd');

const aDate = moment('2020-11-28T04:13:30.000Z');
//clone to not modify the aDate, important !
const previousDay = aDate.clone().add(-1, 'day');
const fourDaysLater = aDate.clone().add(4, 'days');

Here is how you can use it
function drawChart(dayData) {
  //You have a date somewhere, you use it with getDayWeek but I don't see it in the provided code
  //let's put something
  var date = new Date('2020-11-19T04:15:00.000Z');

  const dataTable = ["Day", "Battery", { role: "style" } ];

  new Array(7).fill().forEach( (e, i) => {
    const day = moment(date).add(i, 'days').format('ddd');
    //Where does data1,data2, come from?
    const data = '??';
    const color = (i===0) ? '#2196F3' : '#B33D3C';
    dataTable.push([day, data, color])
  });
  var data = google.visualization.arrayToDataTable(dataTable);
}

A: You can add day worth of milli seconds to get next date.

const followingDays = (dateStr) => {
  const msInDay = 24 * 60 * 60 * 1000;
  return new Array(7)
    .fill(0)
    .map((_, i) =>
      new Date(new Date(dateStr).getTime() + i * msInDay).toDateString()
    );
};

console.log(followingDays((new Date().toString())));
console.log(followingDays('2020-11-24'));

redpajama_set_name: RedPajamaStackExchange


of the outstanding contractors/Beneficiary who has not received their payments.
only) For Such contractors/Beneficiaries. Please re-confirm to me the followings.
(2) Phone, fax and mobile #.
3) Company's name, position and address.
4) Age and marital status.


DIANA FLEMING has been a full-time instructor with the Language Arts Division of De Anza College since 2000. She teaches college composition courses, at both Introduction and Intermediate levels; literature courses; and Basic Skills-level writing strategies. Assigned readings tend to focus on themes of ethnic and cultural identities; socio-economic and psychological influences; gender studies; feminist perspectives; and new historical literary critical theory. She also advises and coaches her students on Transfer Student College Application writing responses.
Fleming's various past, and current, involvements include: Adelante Bridge Program Instructor at San Jose City College; APASA (Asian Pacific Staff Association member); English Department Hiring Committee member; Equal Opportunity Hiring Representative for other departments; and Tenure Chairperson roles. Her most recent Professional Development research project centered on identifying effective pedagogy for Multilingual Student Writing, along with studying Basic Mandarin.
When not teaching, Fleming enjoys spending time with her husband and son. Her hobbies and interests include interior design and styling; fashion history and wardrobe styling; film studies; enjoying various cuisines; and of course, reading literature.


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text:
Iceland government faces breakup as coalition partner quits
The turmoil has surfaced just as the coalition was presenting the budget to parliament. Iceland's economy is booming after the government earlier this year dismantled the last of the capital controls that had been in place since 2008.
(Bloomberg) Iceland is again facing a crisis as coalition partners abandoned the Independence Party-led government over a scandal involving the prime minister's father.
The four-lawmaker Bright Future party announced late on Thursday in Reykjavik that it was leaving the three-party coalition less than a year after it was formed. Hours later, the Reform Party, which has eight legislators, called for new elections as soon as possible.
The coalition led by Prime Minister Bjarni Benediktsson's Independence Party has only a one-seat majority in parliament, and was cobbled together earlier this year after long negotiations. Iceland now faces its second snap election in a year after the previous government unraveled when then Prime Minister Sigmundur David Gunnlaugsson became the most high-profile casualty of the so-called Panama Papers.
The Bright Future party accused the premier of committing a "breach of trust" after failing to inform the rest of the government that his father had written a letter vouching for the character of a convicted child molester so he could apply for a legal "clean slate." The issue has been much in the news in Iceland, where convicted paedophiles can apply to have their record expunged.
"That goes against our policies here at Bright Future on transparency and a good way of working," said Ottarr Proppe, head of the Bright Future party.
Protests were called for on Friday in Reykjavik, which has seen raucous street demonstrations both after the country's economic collapse in 2008 and during the Panama Paper's scandal last year. Benediktsson's name also surfaced in the Panama Papers, but he said he had previously disclosed his offshore holdings.
"Our PM, who was in the Panama Papers, has hidden for two months his father's support for a paedophiles clemency," Smari MacCarthy, a Pirate Party member of parliament, said on Twitter. The party surged in 2016's election, winning 14.5% of the vote and 10 seats in parliament.
The Independence Party hadn't issued any statements as of 5 am on Friday. The prime minister's office didn't immediately respond to calls for comment outside normal business hours.
The current coalition was formed in January after two months of negotiations as the Independence Party, which resisted a challenge from the populist Pirate Party to win the most votes in an October 29 snap elections, replaced its traditional coalition party, the Progressive Party, with the two new junior allies. It controls 32 seats in the 63-member parliament.
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text:
4th Brigade Bylaws & Regulations
Member Units
Artillery Units
Cavalry Units
Infantry Units
Navy/Marine Units
4th Brigade Morning Reports
Medical Information Form
Parent Release Form
1) Col. Chuck Munson
2) Lt. Col. Mitch Price
3) Maj. Ed Mandes
4) Maj. Ken Baum
5) Maj. Joey Campbell
6) Maj. Cameron Kempsel
7) Capt. Greg Youtz
8) Sgt. Mjr. Hamilton McElroy
9) Sgt. Mjr. Tim Sorensen
Brigade Level Forms
Company Level Forms
Money & Labels
Soldier's Papers
Drill Instruction
Soldier's Information
Videos - Drill & Instruction
Video: Artillery Drill
Video: Company Drill
Video: Miscellaneous
Video: Musket Drill
Member Units > Artillery Units >
1st Florida Artillery
1st US Artillery
3rd Rhode Island Artillery
3rd US Artillery
(Guest Unit) 7th Michigan
1st BN - 107th Ohio
1st BN - 12th New Jersey
1st BN - 17th Connecticut
1st BN - 25th Ohio
1st BN - 47th New York
2nd BN - 14th Brooklyn
2nd BN - 3rd Maine
USS Ft. Henry
USS Pawnee
1st Florida Artillery, U.S.
Also known as Sheldon's Mule Battery, U.S.
2nd Lt. Scott Anderson (Founder of 1st Florida Artillery)
1st Sgt. Thomas Van Dyke (1st Florida Artillery Battery - Cannon Owner and Chief of the Piece)1stFloridaArtillery@gmail.com
We Are Presently Recruiting!
Location: Sarasota Area and on the Florida Gulf coast (Central Florida)
Information: Florida 1st Battery Light Artillery is an Independent Company, Field, Mounted (Cannoneers being on foot or riding on the team of horses and on the wheeled Limbers and Caissons) Gun Detachment and Platoon. We Field a Model 1841 12 Pounder Mountain-Howitzer cannon. When our Mountain Howitzer is attached to the 4th Brigade, it may be In Battery or may also serve as Infantry Support (as a Battalion Gun) when ordered. in the event of "capture," It serves as Florida 1st Regiment Light Artillery Company B, C.S. (Dykes Company, Florida Light Artillery).
Unit Website: https://sites.google.com/view/1stFloridaArtillery
Unit Facebook Site:
https://www.facebook.com/1stFloridaArtillery/
Unit History:
Raised at the Warrington Outpost of Fort Barrancas, in Escambia Co., Florida during 1863 around the same time as Florida 1st Cavalry, U.S. under the command of Brigadier General Alexander Asboth. Florida 1st Battery Light Artillery, U.S. has a documented history and was also known as Sheldon's Mule Battery, because there were a shortage of horses in Pensacola for this battery and two companies of Cavalry.
This independent battery formed by recruiting local Unionist Volunteers harassed by Secessionist neighbors and Confederate States Conscription deserters from the surrounding areas of Alabama, Georgia and Florida. The reputation of the 1st Cavalry was blemished by a particular 2nd Lt Joseph G. Sanders who commanded Company F manned with notorious "Scalawags" and "Bushwackers," leading to the branding of most Yankees at Pensacola as such or worse by the locals.
It is not known what guns the 1st Florida Artillery actually used. For smooth bores, the Light Artillery mostly used a combination of two Model 1841 12 Pdr. Field Howitzers and either four 6 Pdr. Smooth Bore Guns or four 24 Pdr. Howitzers. Ours, being the light model of the Howitzers, is used to depict the 12 Pdr. Howitzer in action at reenactments regardless of tube length. when on expedition, either the Full Battery or a two gun Section would have been detached with a Cavalry or Infantry Force. In the 1st Florida Artillery's U.S. case, we presume the Field Howitzers were available . The 2nd and 3rd Brigade Columns, based out of Pensacola, were formed from combining the Florida 1st Regiment Cavalry and other Garrison units stationed at Fort Barrancas as a Battalion for offensive operations.
From Dyer's Compendium:
July 21-25, 1864 Expedition from Barrancas toward Pollard, Ala
July 22, 1864 Action, Camp Gonzales
August 13-14, 1864 Expedition from Fort Barrancas
August 29, 1864 Expedition from Barrancas to Marianna
Oct 18, 1864 Skirmish, Pierce's Point near Milton
October 18-31, 1864 Operations in West Florida
Provenance of The Gun:
Approximately 400 - 500 Model 1841 12 Pdr. Mountain Howitzers served in the War Between the States. The Union Fielded the majority of them. the Confederacy Fielded at least 25 made at the Tredegar Iron Works in Richmond and a number of captured pieces.
The 1860's cost was around $150 U.S. and was In production from 1841 through 1870. It was General John Gibbon's (author of the Artillerist's Manual) favorite gun and Generals John Hunt MOrgan, John S. Mosby and Nathan Bedford Forrest used them, the former called them his Bull (Dog) Pups. They were highly mobile as Horse Artillery (all cannoneers rode horses to keep pace with the Cavalry) and were employed as "Flying" Artillery as First promoted by Major Samuel Ringgold with Light 6 Pounders during the Mexican War.
It is believed this particular gun could have been First ordered cast by one of the Grand Army of the Republic (G.A.R.) Posts in Lynn Haven, Fla. (Panama City) circa 1920's as Signal Gun (Ceremonial Salute Firings.
It became forgotten while left in storage for many years.
Upon rediscovery, it eventually came under a Panama City Reenactor's ownership. There is an early photo of a very good resemblance of this gun from a Reenactment of the Battle of Olustee, during what could be the late 1970's or early 1980's. It shows a Confederate Battery a Line of Artillery. Olustee's First Annual Reenactment dates back to 1981. If it was in their possession for up to 23 years, that may have started around 1978.
Longtime Reenactor and Sutler Seventh Owner, R. Scott Anderson, purchased the piece in 2001. During his ownership, It served alongside Florida 7th Infantry Reenacting Company K and was known as 6th Battalion Co. B (Light Artillery). He added his 12 Flag Streamers to the National Flag coming from is Reenacting Infantry Service from 1989 through 1994.
He Fielded it as Florida Independent (Light) Artillery, Co. B, C.S. and latter transferred to over to the 4th Brigade U.S. as 1st Florida Artillery, (Sheldon's Mule Battery). Christened as "Charlotte, The Harlot."
It was recently purchased by Tom & Dawn Spakowski-Van Dyke in February 2019. She has been re-christened as "Esther Mae" for his Dear Mom, an avid Antebellum History Enthusiast.
Tom started reenacting in 1991 as Infantry and has two ancestors who fought in the War in the Infantry from the 10th and 48th Indiana Regiments. Dawn started reenacting in 2008 and has one ancestor that fought as Infantry from the 20th Indiana Regiment and later transferred to 1st U.S. Artillery, Co. K at Fredricksburg and fought at East Cavalry Park, Gettysburg, on Final action of the Third Day.
Gun Facts:
Class I Ordnance for use as Field (Light) Artillery
Replica Smooth Bore Muzzle Loading, Model 1841 12-Pdr. Mountain Howitzer Gun
Full Scale Second Model, 1861 2nd Prairie Carriage on 42" wheels
Full Scale 30.91" Long Cannon Tube Sub-Bored for to 2" .50 (2 Pdr.) diameter size
Maximum Range Shell at 5 Degrees Elevation was 1005 yards
Firing Blank Chargers Simulating the following Ordnance:
Explosive Spherical (hollow iron ball also called Common) Shell packed with gun powder to set off Fires
Explosive Spherical Case (hollow iron ball, also called Shrapnel) Shot packed with 78 .69 cal. anti-personnel lead balls
Canister (round tin can cylinder) packed with 148 .69 cal. anti-personnel lead balls
Howitzers were in Battery in the West, or in pairs as a Section of Horse Artillery for Cavalry. Their usefulness was mobility as in "Flying" (Horse) Artillery for rapid Field movement response.
Their original design was for Mountain Service such as General John Fremont had on his expedition to California. Light enough to pack on Mules and Horses and even to carry up to a roof tops and a Church Building Steeple, which was the idea of one 2nd Lt. Ulysses S. Grant at San Cosmé during the Mexican War. it was also employed in the 1862 Union Victory at Glorieta Pass, New Mexico (called "The Gettysburg of the West").
© Copyright 2014, in compliance with the Digital Millennium Copyright Act of 1998, by the 4th Brigade, US. All rights reserved. Violators of our copyright will be prosecuted to the full extent of the law. For questions, contact irishham97@gmail.com


It is no secret that a lot of the fans of The Chilling Adventures of Sabrina ship the main character with her original mortal boyfriend Harvey despite their breakup! But is the actor who plays him, Ross Lynch, romantically involved with another co-star in real life instead?
During an interview with ET that also included Ross' brother Rocky, he addressed the rumors that he and Sabrina co-star, Jaz Sinclair are dating!
Fans started to be suspicious after seeing a few social media posts from both actors.
But while Lynch did not give a definite yes or no answer to the dating speculations, it sounds like Jaz has already won over his family since he revealed that she'd come over his home before and cooked for his brother Rocky!
'It was all vegetables, which was like, money,' Rocky said.
Furthermore, this might be a case of art imitating reality since the actor teased that in the new season, fans of Sabrina can expect his and Jaz's characters to start a 'new relationship' of sorts.


The 2012 economic data for China's provincial economies has been released, with Tianjin leading the rankings with the fastest GDP (gross domestic product) growth rate, followed by Chongqing and Guizhou.
The National Statistics Bureau of China announced on Monday that the country's GDP had reached 51.93 trillion yuan (US$8.31 trillion) in 2012, up 7.8 percent from 2011, and 0.3 percent higher than expected.
Tianjin was the fastest mover during this period, with its GDP reaching 1.29 trillion yuan (US$206.41 billion) in 2012, an increase of 13.8 percent from the previous year. Chongqing tied Guizhou for second place, with a growth rate of 13.6 percent.
The GDP per capita in Zhejiang, Jiangsu and the Inner Mongolia Autonomous Region broke through the US$10,000 line for the first time in history. Before 2012, only Beijing, Tianjin and Shanghai, three provincial economies, had a GDP per capita exceeding US$10,000, which is considered as a threshold for an economy to be regarded as "developed."
The following are the top 10 fastest-growing provincial economies on the Chinese mainland for 2012.
The 2012 economic data for China's provincial economies has been released, with Tianjin leading the rankings with the fastest GDP growth rate.


text:
home > International > Boarding Principles
To provide a safe structured family environment which nurtures the potential of the individual and increases self-confidence in preparation for a successful future
To promote and encourage academic success
To increase self-confidence through involvement in house activities and effective assimilation in to the WGS Community
To prepare pupils for a successful future after WGS through affective guidance
To instil a sense of responsibility for their own actions, the school and the wider community.
To encourage a sense of family and mutual respect
To increase pupils' understanding of fundamental British values and their English language skills
The Houses
Hazel House is located on the edge of the school grounds behind the Dwight Centre it is a free standing Boarding House that benefits from a quiet safe setting, it also has its own garden. The House is spread over three floors with a Houseparent on the second floor. The Ground floor has a duty office where a staff member can be found. We have a large common room for relaxation and newly refurbished study bedrooms for focussed homework and rest.
Warren House is situated at the far-side of the school fields. It is very much a home, with all the comforts one would expect! Warren is a converted residential house, so it is no wonder it feels so homely. The house is spread over two floors, with the staff office and resident member of staff situated on the first floor. Boys' accommodation can be found on the ground floor, and the girls' can be found the other side of the kitchen on both the ground floor and first floor.
Despite both houses having their own unique character, the community prides itself on the ethos of teamwork. We look after each other and celebrate our successes and face our challenges together.
We relish the opportunity to represent the school even in areas where we might not have obvious strengths. We believe that you grow as people by performing outside of your comfort zone.
Wisbech Grammar School Boarding Principles
The international strategy at Wisbech Grammar School is very much a symbiotic relationship, it allows our school and its community to stretch out its arms across the world to encounter and learn from different countries and cultures, whilst those who join the school as boarders, benefit from the many opportunities available here such as, our nurturing close-knit community and our focus on an all-round education that inspires.
The aim of boarding at Wisbech Grammar School is not to bolt on a large number of international borders but to allow our international students the opportunity to fully integrate within our own community, housing them in bespoke, intimate, boarding settings and homestay accommodation, to really give them a true home from home and a legitimate cultural experience, from which they can progress and grow as both learners and young people.
This also gives our domestic students a true insight into other cultures and their customs, bursting the bubble that can easily form around a small independent school. Letting the world in further opens up our pupils to new uncharted horizons giving them the opportunity to forge strong links with young people in different countries and eventually the chance to travel and benefit from a portfolio of international contacts.
We are truly proud of our school which has established strong, deep roots in Cambridgeshire in its long and celebrated history. We want others to benefit from the full education on offer here and for our community to grow and become stronger and closer as a result.
Sports Challenge and Skill Videos
Elite Sports Programme


That's to say that he was "angry" and stayed angry in the best sense, in the sense that Jesus was angry, and MLK Jr. was angry, and Gandhi and other prophets of love, peace and justice stayed angry.
By all accounts Seeger was gentle and pleasant company. He could be a playful and joy-filled entertainer and a childishly creative man, which made him and his music so appealing to children as well as his elderly peers.
But he stayed angry, in the best sense, for 94 years, in that he refused to accept social injustice, or the sort of Foxy patriotism that is the last refuge of scoundrels, or Christian sentimentalism.
Jesus himself has to be happy to have this gifted and high-minded prophet back Home. But it saddens me to no end that an American original of such rare integrity and high-minded Christian ethics is no longer here to do what prophets do–to comfort the afflicted and afflict the comfortable.
Driving up the mountain passes of the scenic Hummingbird Highway of Belize sometimes reminds me of the time I vacationed driving through the Smoky Mountains of Tennessee, except for Tennessee's considerable lack of palm and coconut trees. Still, I have these Belizean moments when I feel like I'm back in some place like Tennesseee, or Arkansas, or Georgia (come see Mountain Pine Ridge and you'll understand what I mean, Georgians), or even the holy ground where I grew up around Grimes and Washington Counties in Texas with their rolling, pastoral pastures and hills. As terribly foreign as Belize can start to feel to me sometimes–I have these occasional moments of deja vu where I feel I've been taken back to some beautiful part of America I passed through once on some summer vacation or going mobile in a rental car on some newspaper assignment.
The Hummingbird Highay, one of the more scenic and smooth rides in all of mainland Belize.
Way more than just hummingbird birds up in the bush off the Hummingbird; take a pair of binoculars and your camera early in the morning or just as the sun sets and you'll cross a lot of birds off your bird-watching list.
The Hummingbird dead-ends in the coastal town of Dangriga, where the horse is as common a mode of transportation as a four-wheel or two-wheel vehicle.
Or judging from those wheels, DON'T call for a horse cab.
Born outdoors, Jesus was the ultimate outsider, God's rabble-rouser.
He never had any interest in being an "insider" with the powers-that-were. And he certainly had no interest in being the kind of king who lorded over his empire while his servants sat at his feet giving him pedicures and fetching his fine wine and cheese.
Imagine what the three magi, bearing their lavish gifts, must have thought when they finally reached what they expected to be a royal palace. They must have wondered what kind of strange king-to-be the baby in the manger was.
What are we to make of the strange fact that the only crown the new-born king was ever to wear was a crown of thorns?
That's how God, whose ways are not our ways, works.