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

 Matroids are important structures in combinatorics, particularly in
 relation to combinatorial optimization and graph theory,
 see~\cite{lawler:matroids,oxley:book,white:matroids3}.
 With any matroid $M$ there is an associated simplicial complex $\Delta(M)$ given by
 the independent
 sets of $M$. Such simplicial complexes are called \emph{matroid complexes} and are
 known to be \emph{shellable}, that is, the maximal faces are equicardinal and
 can be arranged in a certain order that helps inductive proofs. (We give a full definition of shellability in the next section.)
 One key combinatorial invariant associated with a shellable
 complex is its $h$-vector which encodes information such as, for example,
 its face and Betti numbers. For these reasons
  shellable
 complexes have received much attention,
 see~\cite{bjorner:shellable-cohen,bjorner+wachs:shellable-non-pure-I,bjorner+wachs:shellable-non-pure-II,van-tuyl+villarreal,ziegler}. The concept of shellability
 is also important in theoretical
 computer science as
 the entries of the $h$-vector of a graphic matroid $M(G)$ are the
 coefficients of the $H$-form of the reliability polynomial of the underlying graph $G$,
 see~\cite{chari+colbourn}.

  A non-void set of monomials $\mathcal{M}$ is a \emph{multicomplex}
  if whenever $m\in \mathcal{M}$ and $m'|m$, then $m'\in
  \mathcal{M}$. A finite
  or infinite
  sequence $h$=($h_0$, $h_1$, $\ldots$, $h_d$) of
  integers is called an  \emph{$O$-sequence} if there exists a multicomplex
  containing exactly $h_i$ monomials of degree $i$.
  An $O$-sequence is \emph{pure} if there exists a multicomplex containing $h_i$ monomials of degree $i$ such that all the maximal elements in the multicomplex have the same degree. Properties of pure O-sequences
  are mentioned in Section~\ref{sec:prelim}.

 In 1977, Richard Stanley made the following conjecture linking $h$-vectors of matroid complexes and $O$-sequences~\cite{stanley:cohen-macaulay}, (see also~\cite{stanley:combinatorics-commutative-algebra}).
  \begin{conjecture}\label{conj:Stanley}
   The $h$-vector of a matroid complex is a pure O-sequence.
 \end{conjecture}

No progress was made on this conjecture for some considerable time.
  But in
 1997,
 work of Norman Biggs~\cite{biggs:chip-firing,biggs:tutte-growth} together with~\cite{merino:chip-firing-Tutte} implicitly proved
 Conjecture~\ref{conj:Stanley} for cographic matroids.
 For an explicit exposition see~\cite{merino:chip-firing-matroid-complexes}. More recently, the
 conjecture was proved for
 rank two matroids in~\cite{stokes}, for lattice-path matroids
 in~\cite{schweig}, for cotransversal matroids in~\cite{oh} and most recently for rank three matroids in~\cite{ha+stokes+zanello:O-sequences}.

 A \emph{paving matroid} is one in which all circuits have size at least
 $r(M)$. Interest in paving matroids goes back to 1976 when
 Dominic Welsh~\cite{welsh:matroid} asked if most matroids are paving. This question
 was motivated by numerical results obtained in~\cite{blackburn+crapo+higgs:catalogue}, where a catalogue of all matroids with up to eight elements was presented. The numerical
 data was updated in~\cite{mayhew+royle:9} to include matroids with nine elements, and the results made the problem even
 more intriguing.  More
 recently, the
 authors of~\cite{mayhew+newman+welsh+whittle} conjecture that asymptotically
 almost every matroid is paving, that is, the proportion of $n$-element matroids which are paving tends to one as $n$ tends to infinity.

 In this work we give a proof that coloopless paving matroids satisfy
 Conjecture~\ref{conj:Stanley}. Should paving matroids genuinely form a significant proportion of all matroids, then our
 result will be of a different kind from all the previous work on Conjecture~\ref{conj:Stanley}, as all previous work only considers classes of matroids whose size is insignificant compared with the total number of matroids.

 This article is organized as follows. In Section~\ref{sec:prelim} we
 give definitions and basic properties of matroids, $h$-vectors and $O$-sequences. In the next section we prove Stanley's conjecture for
 paving matroids.
 The direct approach to
 Stanley's conjecture is to attempt to get a good bounds on the number of bases of a paving matroid in terms of its number of elements and rank and on the minimum number of elements in a pure multicomplex of degree $r$ in $d$ indeterminates which contains every monomial of degree $r-1$. This was our original approach to the problem but we were unable to obtain good enough explicit bounds. However, there appear to be some intriguing open questions concerning these problems including potential links with various other well-studied combinatorial objects.
A subclass of paving matroids, namely sparse paving matroids, was introduced by Jerrum in~\cite{jerrum:balanced} and has recently received attention in~\cite{mayhew+royle:9}.
 In Section~\ref{sec:bounds} we obtain a good lower bound for the number of bases of a sparse paving in terms of the rank $r$ and number $n$ of elements. We have examples showing that this bound is tight for infinitely many values of $r$ and $n$.
We then move on to consider bounds on the sizes of pure multicomplexes of degree $r$ in $d$ indeterminates which contains every monomial of degree $r-1$ and conjecture a link with the number of aperiodic binary necklaces.
The last section
 contains our conclusions.

\section{Preliminaries}
\label{sec:prelim}
In this section we introduce some definitions and key properties of shellable complexes
 and matroids. We assume some  familiarity with matroid theory.
 For an excellent exposition of
 shellability of matroid complexes see~\cite{bjorner:shellability} and for matroids
 see~\cite{oxley:book}.

 \subsection{$h$-vectors}

 Let $\Delta$ be a  \emph{simplicial complex} on the vertex set $V$ = $\{
 x_1,\ldots, x_n\}$. Thus,  $\Delta$ is a collection of subsets of $V$
 such that for all $i$, $\{x_i\}\in\Delta$, and if $F\in \Delta$
 and $F'\subseteq F$, then $F'\in \Delta$. The subsets in $\Delta$ are
 called \emph{faces} and the \emph{dimension} of a face with $i+1$ elements is
 $i$. The \emph{dimension} of $\Delta$ is the maximum dimension of a
 face in $\Delta$.  Associated with $\Delta$  we have
 its \emph{face vector}  or \emph{$f$-vector} $(f_0,f_1, \ldots,
 f_d)$,  where
 $f_i$ is the number of faces of size $i$ (or dimension $i-1$) in  $\Delta$.
 The \emph{face enumerator} is the generating function of the entries of the $f$-vector, defined by
 \[
     f_{\Delta}(x) = \sum_{i=0}^{d}{f_i x^{d-i}}.
 \]
 The maximal faces of $\Delta$ are called \emph{facets}.
When all these have the
 same cardinality $\Delta$ is said to be \emph{pure}.
 From now on we will only consider pure $d-1$-dimensional simplicial complexes.

Given a linear ordering $F_1, F_2, \ldots, F_t$ of the facets of a simplicial complex $\Delta$,
let $\Delta_i$ denote the subcomplex generated by the facets $F_1, F_2, \ldots, F_i$, that is, $F\in \Delta_i$ if and only if $F\in \Delta$ and $F\subseteq F_j$ for some $j$ with $1\leq j\leq i$.

For a pure simplicial complex $\Delta$ a \emph{shelling} is a linear order
of the facets $F_1, F_2, \ldots, F_t$ such that, for $2 \leq l
\leq t$,
\[ \{ F: F\subseteq F_l \text{ and } F\in \Delta_{l-1}\}\]
forms a pure $(\dim(\Delta)-1)$-dimensional simplicial complex. A complex is said to be \emph{shellable} if it is pure and admits a shelling.

Define, for $1 \leq l \leq t$, $R(F_l)= \{ x \in F_l\, |\,\ F_l\setminus
x \in \Delta_{l-1} \}$, where here $\Delta_0=\emptyset$. The number
of facets such that $|R(F_l)|=i$ is denoted by
$h_i$ and importantly does not depend on the particular shelling, see~\cite{bjorner:shellability}. The vector $(h_0, h_1, \ldots, h_d)$  is called
the $h$-{\it vector} of $\Delta$.
The \emph{shelling
polynomial} is the generating function of the entries of the $h$-vector, given by
 \[
     h_{\Delta}(x) = \sum_{i=0}^{d}{h_i x^{d-i}}.
 \]
  It is well known, see for example~\cite{bjorner:shellability}, that the
  face enumerator and the shelling polynomial satisfy the relation
  \[
          h_{\Delta}(x+1) =  f_{\Delta}(x)
  \]
  and so the coefficients satisfy
  \begin{align}\nonumber
     f_k &= \sum_{i=0}^{k}{h_i \binom{d-i}{k-i}}\\
\intertext{and}
  \label{eq:hk=fk}
     h_k &= \sum_{i=0}^{k}{(-1)^{i+k}f_i \binom{d-i}{k-i}},
  \end{align}
  for $0\leq k \leq d$.
\subsection{Matroids and their complexes}

 A \emph{matroid}  is an ordered pair $M$=($E$, $\mathcal{I}$) such that $E$
 is a finite set and $\mathcal{I}$ is a collection of subsets of $E$
 satisfying the following three conditions:
 \begin{enumerate}
 \item $\emptyset \in \mathcal{I}$;
 \item if $I\in \mathcal{I}$ and $I'\subseteq I$, then $I'\in
 \mathcal{I}$;
 \item if $I_1$ and $I_2$ are in $\mathcal{I}$ and
 $|I_1|<|I_2|$, then there is an element $e \in I_2\setminus I_1$
 such that $I_1\cup \{e\}\in \mathcal{I}$.
 \end{enumerate}

 Maximal independent sets
 are called \emph{bases} and it follows easily from the conditions above that all bases have the same
 cardinality. This common cardinality is called the \emph{rank} of the matroid and is usually
 denoted by $r(M)$ or just $r$.

 One fundamental example is the class of \emph{uniform matroids}. The
 uniform matroid with rank $r$ and $n$ elements is denoted by $U_{r,n}$. A set of its elements is independent if and only if it has size at most $r$.

    We recall some basic definitions of matroid theory. A
minimal subset $C$ of $E$ that is not independent is called a \emph{circuit}. The
\emph{closure} $\overline{A}$ of a subset $A$ of $E$ is defined by
\[
    \overline{A}=A\cup \{a\,|\, M\text{ has a circuit } C \text{ such that }
                         a\in C\subseteq A\cup\{a\}\}.
\]
  A subset $S$ is \emph{spanning} if $\overline{S}=E$. A subset $H$
  is a \emph{hyperplane} if is a maximal non-spanning set. For all the
  other concepts of matroid theory we refer the reader to Oxley's
  book~\cite{oxley:book}.

 If $M$=($E$, $\mathcal{I}$) is a matroid, the family of all
 independent sets forms a simplicial complex of dimension
 $r(M)-1$, which we denote by $\Delta(M)$. The facets of $\Delta(M)$
 are the  bases of $M$  and therefore $\Delta(M)$ is
 pure. Complexes of this kind are called  {\it matroid complexes}.
 Matroid complexes are known to be shellable, see
 \cite{bjorner:shellability}.

\emph{Loops} of a matroid are circuits of size one and consequently do not belong to any independent set.
Consequently they do not play any role in $\Delta(M)$ and so to investigate Conjecture~\ref{conj:Stanley}, we can
 safely just consider loopless matroids.

 Furthermore, \emph{coloops} of a matroid are elements
 contained in every basis. Equivalently, they belong to every facet of $\Delta(M)$.
 Suppose $M$ is formed from $M'$ by deleting a coloop. Then $r(M)=r(M')-1$ but more pertinently
  if the  $h$-vector of $M$ is $(h_0,
 h_1, \ldots, h_r)$, then the $h$-vector of $M'$ is $(h_0,
 h_1, \ldots, h_r, 0)$. Thus, all the relevant information concerning the $h$-vector of a matroid
 can still be obtained after
 deleting all its coloops. Consequently, for our purposes we only need to consider coloopless matroids.

\subsection{Pure O-sequences}
\label{multi}

  An explicit characterization of O-sequences can be
  found in \cite{stanley:cohen-macaulay}.
  However, a complete characterization is not known for pure
  O-sequences but Hibi~\cite{hibi} has shown that a pure
  O-sequence $(h_0, h_1, \ldots, h_d)$ must satisfy the following
  conditions:
  \begin{align}
           h_0&\leq h_1 \leq \cdots \leq h_{[d/2]}\label{eq:oseq1}\\
\intertext{and}
\label{eq:oseq2}
      h_i&\leq h_{d-i},  \text{ whenever } 0 \leq i \leq [d/2].
   \end{align}
   Hibi also conjectured that the $h$-vector of a matroid complex must
  satisfy inequalities~\eqref{eq:oseq1} and~\eqref{eq:oseq2}.

The following result concerning the $h$-vector of a matroid complex
  is due to Brown and Colbourn~\cite{brown+colbourn}.
    \begin{theorem}\label{th:Brown-Colbourn}
      The $h$-vector of a connected rank-d matroid satisfies the
      following inequalities:
      \begin{equation}
        \label{eq:bc}
          (-1)^j\sum_{i=0}^{j}{ (-b)^{i}h_i} \geq 0,\qquad 0\leq j \leq d,
       \end{equation}
      for any real number $b \geq 1$ with equality possible
      only if b=1.
    \end{theorem}
   This theorem shows that the converse of Stanley's conjecture is not true because the sequence $(1,4,2)$ is a pure $O$-sequence but does not satisfy the conditions of the theorem.

 Later, Chari~\cite{chari:decompositions} gave a stronger result that generalizes Theorem~\ref{th:Brown-Colbourn}
 and solves Hibi's conjecture.
  The fact that the $h$-vector of a coloop free matroid satisfies
  inequalities~\eqref{eq:oseq1}--\eqref{eq:bc} can also be proved~\cite{chari:inequalities}
  using the Tutte polynomial.

\section{Stanley's conjecture for paving matroids}
\label{sec:Stanleyconj}
A paving matroid $M=(E,\mathcal{I})$ is a matroid whose circuits all have size
at least $r(M)$. If $M$ is a rank-$r$ paving matroid, the face vector of $\Delta(M)$
 is easy to compute. Every subset of size $i<r$ is a face of
 $\Delta(M)$ and the  facets are the bases of $M$. Then, we get the
 following result which is implicit in~\cite{bjorner:shellability}.

 \begin{proposition}
  The $h$-vector of a rank-$r$ paving matroid with $n$ elements and
  $b(M)$ bases is $(h_0,\ldots,h_r)$ where $h_k=\binom{n-r+k-1}k$
  for $0\leq k\leq  r-1$ and $h_r=b(M)-\binom{n-1}{r-1}$.
 \end{proposition}
 \begin{proof}
   Using~\eqref{eq:hk=fk} and $f_i=\binom{n}{i}$ for $0\leq i\leq
   r-1$ we see that
  \[
    h_k=\sum_{i=0}^{k}{(-1)^{i+k} \binom{r-i}{k-i}}\binom{n}{i}.
  \]
  for $0\leq k\leq r-1$. Using the identity
  $(-1)^{a}\binom{b}{a}=\binom{a-b-1}{a}$ we  get
  \[
    h_k=\sum_{i=0}^{k}{\binom{k-r-1}{k-i} \binom{n}{i}}.
  \]
 Now using the Vandermonde convolution formula
 $\binom{a+b}{k}=\sum_{i=0}^{k}{\binom{a}{i} \binom{b}{k-i}}$ we get
 \[
   h_k=\binom{n-r+k-1}{k}.
 \]
 Because $\sum_{i=0}^{r}{h_i}=b(M)$, we get
 \[h_r=b(M)-\sum_{i=0}^{r-1}{h_i} = b(M) - \sum_{i=0}^{r-1}{\binom{n-r+i-1}{i}} = b(M) - \binom{n-1}{r-1}.\]
 \end{proof}

 The idea for proving that the $h$-vector of a coloopless
 paving matroid is the O-sequence of a pure multicomplex is simple. We
 define the multicomplex
 $\mathcal{M}_{r,d}$ to be the pure multicomplex in which the maximal
 elements are all monomials of
 degree
 $r$ in $d$ indeterminates $z_1,\ldots, z_{d}$. This multicomplex
 has $O$-sequence $(h_0,\ldots,h_{r})$, where $h_k=\binom{d+k-1}k$.

 Now, define the function
\[
      f(r,d)=\min\{ h_r\,|\, (h_0,\ldots h_r)\ \text{is the pure O-sequence
      of} \ \mathcal{M}\supset \mathcal{M}_{r-1,d}\}.
\]
This means that $f(r,d)$ is the minimum number of monomials of degree
$r$ in a pure multicomplex of degree $r$ which contains every monomial
of degree $r-1$ in the $d$ indeterminates $z_1,\ldots,z_d$.
So for any positive integers $d$ and $r$, if $h_k = \binom{d+k-1}k$
for $0\leq k \leq r-1$ and $f(r,d) \leq h_r \leq \binom{d+r-1}{r}$,
the sequence $(h_0,h_1,\ldots,h_r)$ is a pure $O$-sequence.

If $M$ is a paving matroid with $n$ elements and rank $r$, then by
taking $d=n-r$, we see that the $h$-vector of $M$ satisfies $h_k =
\binom{d+k-1}k$ for $0\leq k \leq r-1$. To prove Stanley's conjecture
for paving matroids, it will be sufficient to prove that $f(r,d) \leq
h_r \leq \binom{d+r-1}{r}$ or equivalently
\[f(r,n-r) \leq b(M) - \binom{n-1}{r-1} \leq \binom{n-1}{r}.\]
The second inequality is trivial since $b(M)\leq \binom nr$, so we
focus on the first inequality.

Some initial
values of $f$ are easy to get.
\begin{lemma}\label{lem:initial_values_f}
  For $r\geq 1$ and $d\geq 1$ we have that $f(1,d)=1$, $f(2,d)=\lceil
  d/2 \rceil$, $f(r,1)=1$ and $f(r,2)=\lceil r/2 \rceil$.
 \end{lemma}

 \begin{lemma}\label{lem:recursion_f(p,d)}
   $f(r,d)\leq f(r,d-1)+f(r-1,d)$.
 \end{lemma}
 \begin{proof}
Let $\mathcal{M}'$ be a multicomplex in indeterminates
$z_1,\ldots,z_{d-1}$ containing $\mathcal{M}_{r-1,d-1}$, having
$h$-vector $(h_0',\ldots,h_r')$ satisfying $h_r'=f(r,d-1)$.
Let $\mathcal{M}''$ be a multicomplex in indeterminates
$z_1,\ldots,z_d$ containing $\mathcal{M}_{r-2,d}$, having $h$-vector
$(h_0'',\ldots,h_{r-1}'')$ satisfying $h_r''=f(r-1,d)$.

Consider the multicomplex $\mathcal{M}$ that is the
union of $\mathcal{M}'$ and \[z_d\mathcal{M}''=\{z_d\,m|\,m\in
    \mathcal{M}''\}.\] Then, $\mathcal{M}$ contains
 all
    the monomials over $z_1,\ldots, z_{d-1}$ of degree at most $r-1$ and
    all the monomials over $z_1,\ldots, z_{d}$ of degree at most $r-1$
    where $z_d$ has degree at least 1. These are precisely all the
    monomials over  $z_1,\ldots, z_{d}$ of degree at most
    $r-1$. Therefore $\mathcal{M}$ contains $\mathcal{M}_{r-1,d}$.

    It remains to prove that $\mathcal{M}$ is a multicomplex. Let
    $m\in \mathcal{M}$ and $m'|m$. Then $m'$ is a monomial in
    indeterminates $z_1,\ldots,z_d$ and either $m'=m$ or $m'$ has
    degree at most $r-1$. By using the previous part of the proof, in
    either case we obtain that $m \in \mathcal {M}$.

    Finally, the O-sequence of  $\mathcal{M}$  is $(h'_0,
    h'_1+h''_0,\ldots, h'_r+h''_{r-1})$.

  \end{proof}

 Let $\mathcal{P}_{r,n}$ be the class of coloopless, loopless rank-$r$ paving
 matroids on $n$ elements. We define
 \[
      g(r,n)=\min\Big\{ b(M)-\binom{n-1}{r-1}\,|\, M\in \mathcal{P}_{r,n}\Big\}.
\]

 Observe that $g(r,n)$ equals the minimum value of $h_r$ among all $h$-vectors
 of matroids in $\mathcal{P}_{r,n}$. Thus, to prove
 Stanley's conjecture for paving matroids is enough to show that
 $g(r,n)\geq f(r,n-r)$.

  \begin{lemma}\label{lem:rank_1}
   For all $n\geq 1$, $g(1,n)\geq f(1,n-1)$.
 \end{lemma}
 \begin{proof}
    Up to isomorphism, the only matroid, in $\mathcal{P}_{1,n}$ is
   $U_{1,n}$, thus $g(1,n)=n-1$ and $f(1,n-1)=1$.
  \end{proof}

  \begin{lemma}\label{lem:rank_2}
    For all $n\geq 2$, $g(2,n)\geq f(2,n-2)$.
 \end{lemma}
 \begin{proof}
   It is enough to prove that for any loopless and coloopless rank-2
   paving matroid $M$ with $n$ elements, $b(M)\geq (n-1)+ \lceil (n-2)/2
   \rceil $. If every  element is in at least 3 bases, then $b(M)\geq
   3n/2> 3(n-1)/2\geq (n-1)+ \lceil (n-2)/2 \rceil $.

  Suppose that $M$ has an element $e$ in at most 2 bases.
  If the number of parallel classes in $M$ is at least 4, then every
  element is in at least 3 bases. Thus, $M$ has at most 3 parallel
  classes. If $M$ has two parallel classes, it is not possible that
  both have size at
  least 3, thus, there is a parallel classes of size 2. Therefore, $M$ is
  $U_{1,n-2}\oplus U_{1,2}$, where $n\geq 4$ as $M$ is coloopless, and $b(M)=2(n-2)\geq (n-1)+ \lceil (n-2)/2
  \rceil$, with equality when $n=4$.

  If there are three parallel classes, it is not possible that two
  have size at least 2, or else every element is in at least three bases. Thus there are two parallel classes of size one
  and one of size $n-2$. Therefore, $M$ is isomorphic to
  $U_{1,n-1}\oplus_2 U_{2,3}$, that is, the graphic matroid of a triangle with one edge replaced by $n-2$ parallel edges, and $b(M)=2(n-2)+1\geq 3(n-1)/2\geq
  (n-1)+ \lceil (n-2)/2 \rceil$ with equality when $n=3$ and $M$ is
  $U_{2,3}$.
  \end{proof}

 We use the following result from~\cite{chavez+merino+noble+ibanez}.
   \begin{lemma}\label{lem:2-stretching}
    Let $M$ be a rank-r coloopless paving matroid. If for every element $e$
    of $M$, $M\setminus e$ has a coloop, then one of the following
    three cases happens.
    \begin{enumerate}
    \item $M$ is isomorphic to $U_{r,r+1}$, $r\geq 1$.
    \item $M$ is the 2-stretching of a uniform matroid
       $U_{s,s+2}$, for some $s\geq 1$.
    \item $M$ is isomorphic to $U_{1,2}\oplus
    U_{1,2}$.
    \end{enumerate}
   \end{lemma}

  \begin{lemma}
    Let $M$ be a rank-r coloopless paving matroid with $n$
    elements. If for every element $e$ of $M$, $M\setminus e$ has a
    coloop, then $b(M)- \binom{n-1}{r-1}\geq f(r,n-r)$
 \end{lemma}
 \begin{proof}
    It follows from the previous lemma that we just have to check
    three cases. If $M\cong U_{r,r+1}$, $b(M)=r+1=\binom r{r-1}+ f(r,1)$. In this case we have equality.

    If the matroid $M$  is the
   2-stretching of $U_{s,s+2}$, it has rank $2s+2$, $2s+4$ elements and
   $2(s+2)(s+1)$  bases, thus $b(M)-\binom{2s+3}{2s+1}=s+1$. On the
   other hand, $f(2s+2,2)=s+1$, and we have equality.

Finally, if
    $M\cong U_{1,2}\oplus U_{1,2}$, then $M$ has rank 2 and has
    already been considered in Lemma~\ref{lem:rank_2}.
  \end{proof}

 \begin{theorem}\label{thm:main}
   For $r\leq n$ we have $g(r,n)\geq f(r,n-r)$.
 \end{theorem}
  \begin{proof}
    We prove the statement by induction on $r+n$. If
    $r\leq 2$ and $n\geq r$, the result follows by Lemmas~\ref{lem:rank_1}
    and~\ref{lem:rank_2}. Suppose that the statement is true for all $r'$ and $n'$ with $r'+n'<r+n$.

    Let $n\geq r>2$ and $M$ be a matroid in $\mathcal{P}_{r,n}$ such that
    $b(M)=\binom{n-1}{r-1}+g(r,n)$. Suppose that $M\setminus e$ has
    no coloops for some $e\in E(M)$. Then
    \begin{align*}
       g(r,n)&=b(M)- \binom{n-1}{r-1}
             =b(M\setminus e)-\binom{n-2}{r-1}+b(M/e)-\binom{n-2}{r-2}\\
             &\geq g(r,n-1)+g(r-1,n-1)\geq f(r,n-r-1)+f(r-1, n-r)\\&\geq f(r,n-r).
    \end{align*}
     If $M$ has no such element $e$ then the result follows by
     Lemma~\ref{lem:2-stretching}.
  \end{proof}

 \begin{corollary}\label{Stanley_Paving}
     The $h$-vector of the matroid complex of a paving matroid is a
     pure O-sequence.
 \end{corollary}

\section{Bounds on the number of bases of a sparse paving matroid}
One intriguing problem is to determine more about the functions $f$ and $g$ from the previous section. This appears to be a rather hard problem, in particular we have not been able to find tight bounds on the number of bases of a paving matroid in terms of its rank and number of elements. In this section we find a tight bound for the number of bases for a subclass of paving matroids, namely the sparse paving matroids.

We will require the following result on paving matroids which is an Exercise in~\cite{oxley:book} (Page~132, Exercise~8).
 \begin{proposition}\label{paving_closed_minor}
   Paving matroids are closed under minors. Moreover a matroid $M$ is paving if and only if it does not contain the matroid
$U_{2,2}\oplus U_{0,1}$ as a minor.
 \end{proposition}

Sparse paving matroids were introduced by Jerrum in~\cite{jerrum:balanced,mayhew+royle:9}.
A rank-$r$ matroid $M$ is {\em sparse paving}  if $M$ is paving and
  for every  pair of cycles $C_1$ and $C_2$ of size $r$ we have
  $|C_1\bigtriangleup C_2|>2$. For example, all uniform matroids are
  sparse paving matroids. The following lemma is straightforward.

 \begin{lemma}\label{sparse_rank_1}
  If $M$ is a sparse paving matroid with rank at most 1 and $n$
  elements, then $M$ is isomorphic to $U_{1,n}$, $U_{1,n-1}\oplus
  U_{0,1}$ or $U_{0,n}$.
 \end{lemma}

 \begin{theorem}\label{sparse_circuit_hyperplane}
   Let $M$ be a paving matroid of rank $r\geq 2$. Then $M$ is
   sparse paving if and only if all the hyperplanes of $M$ have size $r$
   or $r-1$ and the hyperplanes of size
   $r$ are  precisely its circuits of size $r$.
 \end{theorem}
 \begin{proof}
    For the forward implication let $H$ be a hyperplane of $M$ and $I$
    be a maximal independent set
    contained in $H$. If there are two elements $e\neq f$ in
    $H\setminus I$, then
    $C_1=I\cup\{e\}$ and $C_2=I\cup\{f\}$ are circuits of size $r$
    but $|C_1\bigtriangleup C_2|= 2$, contrary to the assumption that $M$
    is sparse paving. Thus, any hyperplane has size either $r-1$ or
    $r$ and in the case that $H$ has size $r$ it will also
    be a circuit. Let $C$ be a circuit of $M$ of size $r$. Then
    since $C$ has rank $r-1$, $\overline{C}$, the closure of $C$, is
    a hyperplane. By the previous argument this hyperplane has size
    $r$ and as $C\subseteq
    \overline{C}$, we conclude that $C$ is a hyperplane.

   To prove the converse, we take two circuits $C_1$ and $C_2$ of size
   $r$ in $M$. Then $I=C_1\cap C_2$ is an independent set and because
   $I$ is the intersection of two hyperplanes, its rank is at most
   $r-2$. So, $|C_1\cap C_2| \leq r-2$ and  $|C_1\bigtriangleup C_2|>
   2$.
 \end{proof}
The following result appears to be (recent) folklore but we are unable to find a reference.
 \begin{theorem}\label{sparse_closed_duality}
   If $M$ is an $n$-element sparse paving matroid, then $M^{*}$ is
   also sparse paving.
 \end{theorem}
 \begin{proof}
   If $M$ has rank at most 1, it follows by Lemma~\ref{sparse_rank_1}
   that $M^{*}$ is isomorphic to $U_{n-1,n}$, or $U_{n-2,n-1}\oplus
   U_{1,1}$ or $U_{n,n}$. In each case $M^{*}$ is sparse paving.

   Let us suppose that $M$ has rank $2\leq r \leq n-2$. By duality,
   $C$ is a circuit of a matroid $N$ over $E$ if and only if
   $E\setminus C$ is a hyperplane of $N^{*}$.
   From
   Theorem~\ref{sparse_circuit_hyperplane} it follows that all the
   hyperplanes of $M$ have size $r$ or $r+1$. Consequently all the circuits of $M^{*}$ have size $n-r$ or $n-r-1$ and so $M^{*}$ is paving.

   Furthermore all the hyperplanes of $M^{*}$ have size $n-r$ or $n-r-1$ and since hyperplanes and circuits of $M$ of size $r$ coincide, hyperplanes and circuits of $M^{*}$ also coincide and so by Theorem~\ref{sparse_circuit_hyperplane}, $M^{*}$ is sparse paving.

   Finally, if the rank of $M$ is $n-1$ or $n$, then $M$ and $M^{*}$ are
   uniform matroids and the result follows.
 \end{proof}

The next result was first proved by Jerrum~\cite{jerrum:balanced}. It follows immediately from Theorem~\ref{sparse_closed_duality} and the fact that the collection of circuits of
   $M\setminus e$ is the collection of circuits of $M$ that do not contain $e$.
 \begin{theorem}
   Sparse paving matroids are closed under minors
 \end{theorem}

  \begin{theorem}
      A matroid $M$ is sparse paving if and only if it does not have
   $U_{2,2}\oplus U_{0,1}$ nor $U_{0,2}\oplus U_{1,1}$ as  minors.
  \end{theorem}
  \begin{proof}
   By Theorem~\ref{paving_closed_minor}, it is enough to prove that a
   paving matroid $M$ is sparse if and only if $M$ does not contain
   $U_{0,2}\oplus U_{1,1}$ as a minor.

   If a rank-$r$ paving matroid $M$ contains $U_{0,2}\oplus U_{1,1}$ as a minor,
   then $M^{*}$ contains $U_{2,2}\oplus U_{0,1}$ as a minor and by Proposition~\ref{paving_closed_minor} it is not
   paving. Thus $M$ cannot be sparse paving by
   Theorem~\ref{sparse_closed_duality}.

   Suppose $M$ is a rank-$r$ paving matroid with $n$ elements that is not
   sparse. All hyperplanes of $M$ of size $r$ must be circuits because every set of size $r-1$ is independent.
   Consequently there must exist
   a hyperplane $H$ of size at least $r+1$.
   Let $I$ be a maximal indpendent set in $H$ and let $\{e,f\}\in H\setminus I$.
   Now let $g\not\in
   H$. If we delete the elements in $E\setminus (H\cup\{g\})$ and
   contract the elements in $H\setminus \{e,f\}$ we get a
   $U_{0,2}\oplus U_{1,1}$ minor.
  \end{proof}

 In order to get more properties of sparse paving matroids, we need
 the following definition from~\cite{oxley:book}. Given integer $k>1$ and  $m>0$, a collection  $\mathcal{T}=\{T_1,\ldots,T_k\}$ of subsets
 of a set $E$, such that each member of $\mathcal{T}$ has at least $m$
 elements and each $m$-element subset of $E$ is contained in a unique
 member of  $\mathcal{T}$, is called an $m$-partition of $E$. The
 following proposition is also from~\cite{oxley:book}.

 \begin{proposition}
   If $\mathcal{T}$ is an $m$-partition of $E$, then $\mathcal{T}$ is
   the set of hyperplanes of a paving matroid of rank $m+1$ on
   $E$. Moreover, for $r\geq 2$, the set of hyperplanes of every
   rank-$r$ paving matroid on $E$ is an $(r-1)$-partition of $E$.
 \end{proposition}

 Thus the collection of hyperplanes of a sparse paving matroid $M$ of rank $r\geq 2$ are the circuits of size $r$ together with
 the independent sets of size $r-1$ not contained in any circuit of
 size $r$. Also, because the hyperplanes of $M$ form an
 $(r-1)$-partition, any  subset $A$ of size $r-1$, that is not a
 hyperplane, (so $A$ is an independent set contained in some circuit of size $r$) is  contained in a  unique circuit of size $r$.

Any Steiner system $S(r-1,r,n)$ corresponds to a sparse paving matroid
by taking the bases to be all sets of size $r$ not appearing as blocks of the Steiner system.
As the number of blocks in
 a $S(r-1,r,n)$ is $\frac{1}{r}\binom n {r-1}$ we see that the number
of bases of the corresponding sparse paving matroid is
$\binom n r-\frac{1}{r}\binom n
{r-1}$ = $\frac{n-r}{r}\binom n{r-1}$. The next result shows that this is a lower bound for the number of bases of a sparse paving matroid.  Because the Steiner systems $S(2,3,6p+1)$,  $S(2,3,6p+3)$
  (see~\cite{kirkman}) and  $S(3,4,6p+2)$,  $S(3,4,6p+4)$
  (see~\cite{hanani}) exist
  for all $p$,
  there is an infinite number of matroids that achieve our bound.

  \begin{theorem}
  Let $M$ be a rank-$r$ matroid with $n$ elements and $r\geq 1$.
  If $M$ is a sparse paving matroid then it has at least
  $\frac{n-r}{r}\binom n{r-1}$ bases.
  \end{theorem}
   \begin{proof}
    If $r=1$, $M$ is isomorphic to either $U_{1,n}$, $U_{1,n-1}\oplus
  U_{0,1}$ by Lemma~\ref{sparse_rank_1}. Both of these matroids have at least $n-1$ bases.

  Let us suppose that $r\geq 2$. Because $M$ is paving, every subset
  of size $r-1$ is independent and, because it is sparse, the remarks preceding the theorem imply that any
  set of size $r-1$ is in at most one circuit of size $r$. Now, form
  the bipartite graph of bases and
  independent sets of size $r-1$. That is, the vertices are the
  independent sets of sizes $r$ or $r-1$ and there is an edge $(B,I)$
  if and only if the base $B$ contains the independent set  $I$. The
  degree of any independent set $I$ of size $r-1$ is at least $n-r$.

  So the number of edges in the bipartite graph is at least
  $(n-r) \binom n{r-1}$. As the degree of any basis $B$ in
  this graph
  is $r$, the result follows.
   \end{proof}

 Many invariants that are usually difficult to compute for a general
 matroid are easy for sparse paving matroids. For example, observe
 that if $M$ is sparse paving, all subsets of size $k<r$ are independent,
 and all subsets of size $k>r$ are spanning. On the other hand the
 subsets of size $r$ are either bases or circuit--hyperplanes. Thus,
 the Tutte polynomial of a rank-$r$ sparse matroid $M$ with $n$
 elements and $\lambda$ hyperplanes is
 \[
   T_{M} (x,y)= \sum_{i=0}^{r-1}\binom ni(x-1)^{r-i}+
                           \binom nr + \lambda(xy-x-y)+
                           \sum_{i=r+1}^{n}\binom ni(y-1)^{i-r}.
  \]

\section{Bounds for number of bases of paving matroids and sizes of multicomplexes}
\label{sec:bounds}

 In the previous section we gave a tight lower bound for the
 number of bases of a sparse paving matroid. Such a lower bound is a
 more difficult to obtain in the case of paving matroids. In
 order to find a bound for the number of bases we investigate further
  the functions $f(p,d)$ and $g(p,n)$ defined in Section~\ref{sec:Stanleyconj}.

 \subsection{The function f(r,d)}

 We define two families of graphs. First, we define the graph
 $G_{r,d}$ to have one vertex corresponding to each monomial of degree $r$ over $d$ variables and have an edge
 $\{m,m'\}$ if and only if there exist distinct variables $x$ and $y$
 such that $m'=\frac{m}{x}y$. The second family is similar. We
 define $TG_{r,d}$ to have the same vertex set as $G_{r,d}$ and have an edge
 $\{m,m'\}$ if and only if there exist different variables $x$ and $y$
 such that $m'=\frac{m}{y}x$ or there exists a variable $y$ such that
 $m'=\frac{m}{y}$.

 Clearly, $G_{r,i}$ is an induced subgraph of
 $TG_{r,d}$ for all $0\leq i\leq d$.
 Recall that a set $U$ of vertices dominates a set $U'$ of vertices in a graph
 if every vertex in $U'\setminus U$ is adjacent to a vertex in $U$.
 The problem of finding $f(r,d)$
 can be translated to the problem of finding the vertex subset of
 $G_{r,d}$ of minimum size that dominates the
 vertex set of
 $G_{r-1,d}$.

 For this purpose we define the {\it standard colouring} $C_{d}$ of
 $G_{r,d}$.  Let us suppose the $d$  variables are $\{x_0,\ldots,
 x_{d-1}\}$. To each variable $x_i$ we associate the colour
 $\varrho_d(X_i)=i \bmod d$ and then  we extend this colouring linearly to all
 monomials, that is, for
 a monomial $m=x_{0}^{t_{0}}\cdots x_{d-1}^{t_{d-1}}$, the value of
 $\varrho_d(m)$ is $0\,t_{0}+\ldots+(d-1)t_{d-1} \bmod d$.

 \begin{lemma}
     The standard colouring $\varrho_d$ is a proper colouring and
     $\chi(G_{r,d})\leq d$.
 \end{lemma}
 \begin{proof}
     If $(m,m')$ is an edge of
        $G_{r,d}$, then there exist $i\neq j$ such that
        $m'=\frac{m}{x_{i}}x_{j}$. So, $\varrho_d(m)-\varrho_d(m')=i-j\not\equiv 0 \pmod
        d$. Thus, $m$ and $m'$ receive different colours and $\varrho_d$ is a
        proper $d$-colouring of $G_{r,d}$.
\end{proof}

  \begin{proposition}
     The chromatic number $\chi(G_{r,d})$ equals the clique
     number  $\omega(G_{r,d})$ and both equal $d$.
 \end{proposition}
 \begin{proof}
         From the previous lemma we know that  $\chi(G_{r,d})\leq d$.
        Clearly the vertices in $\{x_{0}^{r},
        x_{0}^{r-1}x_{1},\ldots, x_{0}^{r-1}x_{d-1}\}$ form a
        clique. Thus $\omega(G_{r,d})\geq d$. But for any graph $G$ we
        have that $\omega(G)\leq \chi(G)$ and the result follows.
  \end{proof}

 Observe that in the previous proof we show that $x_{0}^{r}$ is in an
 $d$-clique. Actually, any monomial in   $G_{r,d}$ is in as many  cliques of
 size $d$ as the number of different variables in the monomial. That is,
 if $x_i\,|\,m$, then the vertices $\{\frac{m}{x_i} x_0,
        \ldots, \frac{m}{x_i} x_{d-1}\}$ form a clique. Thus, any
 colour class of a $d$-colouring of $G_{r,d}$ dominates $V(G_{r,d})$. So any colour class of a $d$-colouring of $G_{r,d}$ is a dominating independent set and
 thus, it is a maximal independent set and a minimal dominating set.

 Another important observation is that a colour class of
 a $d$-colouring of $G_{r,d}$ dominates
 the vertex subset $V(G_{r-1,d})$ in $TG_{r,d}$. This is because the neighbours of a
 monomial $m$ of rank $r-1$ in $V(G_{p,d})$ are $\{m\,x_0, \ldots, m\,x_{d-1}\}$ and
 form a $d$-clique. So they must intersect each colour class of a $d$-colouring of $G_{r,d}$.

 We now define the function $\overline{f}(r,d)$ to be the minimum size of a
 chromatic class in the standard coloring $\varrho_d$ of $G_{r,d}$. The previous
 paragraph proves the following.

 \begin{proposition}\label{ff_bounds_f}
     For all $r\leq d$, we have $f(r,d)\leq \overline{f}(r,d)$.
 \end{proposition}

  Now, it is easy to give an upper bound for $f(r,d)$.

\begin{proposition}
   For all $r\leq d$, we have that $f(r,d)\leq \binom{r+d-1}{d-1}/d$.
 \end{proposition}
\begin{proof}
  $\binom{p+d-1}{d-1}/d$ is the average size of a colour class in a
  $d$ colouring of $G_{r,d}$.
\end{proof}

 While trying to find a formula for $\overline{f}(r,d)$, our
 computations appeared to point to the number of aperiodic  necklaces with $r$
 black beads and $d$ white
 beads, also known as the number of binary Lyndon words of length $r+d$
 and density $r$.  \emph{Binary necklaces} or  \emph{\it necklaces
 of beads with colours black and  white} are circular sequences of 0's
 and 1's, where two sequences obtained by a rotation are considered
 the same. That is, the necklaces of length $n$ are the orbits of the
 action of the  cyclic group $C_n$ on circular
 sequences of 0's and 1's of length $n$. A necklace of length $n$ is
 called {\it aperiodic} if the orbit has size $n$.

 The number of aperiodic necklaces with $n$ beads, $r$
 black and $d$   white,  is
 \[
     L_2(r,d)=\frac{1}{r+d}\sum_{k|(r+d,r)} \mu(k)\binom{(r+d)/k}{r/k},
 \]
 where $(a,b)$ denotes the greatest common divisor of  the integers
 $a$ and $b$ and $\mu$ is the
 classical M\"obius function. This formula is well known and is a typical
 example of the M\"obius inversion formula, see~\cite{van-lint+wilson:book}. In particular,
 note that when $d$ and $r$ are coprimes, the formula simplifies to $ \binom np/n$= $\binom{n-1}{p-1}/p$=$\binom{n-1}{d-1}/d$.

 \begin{theorem}
      If $n$ and $r$ are coprime, then $\overline{f}(r,n-r)$ equals
      the number of aperiodic necklaces of $n$ beads, $r$ black and
      $d=n-r$ white.
 \end{theorem}
 \begin{proof}
   Consider $\varphi$ the action of the cyclic group $C_d$ over
   $G_{r,d}$ given by
   \[\varphi(x_{0}^{t_{0}}\cdots x_{d-1}^{t_{d-1}})
   = x_{1}^{t_{0}}\cdots x_{d-1}^{t_{d-2}}x_{0}^{t_{d-1}}.\] The
   orbits of this action correspond to necklaces with $r$ black beads
   and $d$ white beads. Variables correspond to white beads and to the
   right of the black bead corresponding to  $x_i$ we place as many
   black beads as the exponent of $x_i$, for $0\leq i\leq d-1$. The
   orbits of size $p+r=n$ correspond to aperiodic necklaces.

   Let us see the effect of $\varphi$ on the standard colouring
   $\varrho$, that is, we want to find  $\varrho(\varphi(m))$ for a
   monomial $m$=$x_{0}^{t_{0}}\cdots x_{d-1}^{t_{d-1}}$. We have
   $\varrho(\varphi(m))-\varphi(m)\equiv r \mod d$. Thus, every orbit
   has size $d$ and all the monomials in the orbit have different
   colours. We conclude that in this case the number of aperiodic
   necklaces equals  the common size of any colour class in the
   standard colouring $\varrho$ of $G_{r,d}$.
 \end{proof}

  \begin{conjecture}\label{conjecture}
       $f(r,d)=\overline{f}(r,d)= L_2(r,d)$.
  \end{conjecture}

 Notice that if $I'$ is a set
 of monomials of size $f(r,d)$ which dominates the vertices in
 $G_{r-1,d}$, it is a dominating set in $G_{r,d}$. This is because, if
 $m\in V(G_{r,d})$, then for some $x_i$ the monomial $m'=m/x_i$ is in
 $V(G_{r-1,d})$. But the set of neigbours of $m'$ in $V(G_{p,d})$
 is $S=\{m'\,x_0, \ldots, m'\,x_{d-1}\}$ and $m\in
 S$. As an element $m''$ of $I'$ has to be in $S$ and $S$ induces a
 complete graph, we conclude that $m''$ and $m$ are adjacent.

\subsection{The function g(p,d)}

 For a connected paving matroid we can use the Brown-Colbourn
 Theorem~\ref{th:Brown-Colbourn}
 mentioned earlier to bound $h_r$ for $r\geq 1$ from below by
 \[
    S(r,n)= (-1)^{r-1}\sum_{i=0}^{r-1} {(-1)^{i}\binom{n-r+i-1}i}.
 \]
 A few values of $S$ are given by the following.
 \begin{proposition}\mbox{ }
 \begin{itemize}
  \item $S(1,n)=1$ for all $n\geq 1$.
  \item $S(2,n)=n-3$ for all $n\geq 2$.
  \item $S(n,n)=(-1)^{n-1}$ for all $n\geq 1$.
  \item $S(n-1,n)= n-1 \mod 2$ for $n\geq 2$.
  \item $S(n-2,n)=\lfloor \frac{n-1}{2} \rfloor$
  \end{itemize}
  \end{proposition}
  \begin{proof}\mbox{ }
    \begin{itemize}
  \item $S(1,n)=(-1)^0 \binom{n-1}0=1$ for all $n\geq 1$.
  \item $S(2,n)$ = $(-1) ( \binom{n-3}0 - \binom{n-2}1)$ = $n-3$ for all $n\geq 2$.
  \item $S(n,n)$ = $(-1)^{n-1}\sum_{i=0}^{n-1} (-1)^i\binom{i-1}i$
  = $(-1)^{n-1}$ for all $n\geq 1$. Here we adopt the usual convention that $\binom a0=1$ for
  all $a$ and $\binom ab=0$ for all integers $a$ and $b$
  such that $b>a$ and $b> 0$.
  \item $S(n-1,n)$ = $(-1)^{n-2}\sum_{i=0}^{n-2} {(-1)^i \binom ii}$ = $(-1)^{n-2}\sum_{i=0}^{n-2}{(-1)^i}$ that is 1 if $n$ is even
  and 0 otherwise.
  \item  $S(n-2,n)$ = $(-1)^{n-3}\sum_{i=0}^{n-3} {(-1)^i \binom{i+1}i}$ = $(-1)^{n-3}\sum_{i=0}^{n-3}{(-1)^i (i+1)}$. This is
  $(-1)^{n-3}(-(n-1)/2)=(n-2)/2$ if $n-2$ is even and $(n-1)/2$ if
  $n-2$ is odd.
  \end{itemize}
  \end{proof}
  The sequence $\{S(r,n)\}$ has the same  recursion  as the binomial
  coefficients.
 \begin{theorem}
  For $n\geq r\geq 1$
   \[
      S(r+1,n+1)=S(r+1,n)+S(r,n)
   \]
 \end{theorem}
 \begin{proof}
   This follows directly by the Pascal--Stifel's formula $\binom{n+1}{r+1}$= $\binom n{r+1}+\binom nr$.
  \end{proof}

This result is enough to show that the integer sequence $\{S(r,n)\}$
 is sequence A108561
 in \cite{sloane:online}, as both satisfy the same recurrence and the
 same boundary conditions.

  How does $S(r,n)$
 compare with $f(r,n-r)$? We can prove the following.
 \begin{theorem}
   For $n\geq 3$ and $r\leq n-2$ we have that $f(r,n-r)\leq S(r,n)$
  \end{theorem}
  \begin{proof}
    For $r=1$, $f(1,n-1)=1=S(1,n)$ for all $n$. For $r=n-2$, we have
    $f(n-2,2)=\lceil \frac{n-2}{2} \rceil$ = $\lfloor
    \frac{n-1}{2} \rfloor=
    S(n-2,n)$. By Lemma~\ref{lem:recursion_f(p,d)}, $f(r,n-r)\leq
    f(r,n-r-1)+f(r-1,n-r)$. Using induction this is at most
    $S(r, n-1)+S(r-1,n-1)$ which equals $S(r,n)$.
  \end{proof}

A coloopless paving matroid that is not connected must have rank one, so the previous result implies that $f(r,n-r)\leq
 g(r,n)$ for $n\geq 3$ and $2\leq r\leq n-2$.
 Thus, we have an alternative proof of
 Corollary~\ref{Stanley_Paving} because it is easy to check the inequality for the remaining values of $r$ and $n$.

  \section{Conclusion}

  We have proved Stanley's conjecture for paving matroids. This adds another
  case to the stream of result that prove the conjecture for a
  particular family of matroids~\cite{ha+stokes+zanello:O-sequences,merino:chip-firing-matroid-complexes,oh,schweig,stokes}. However,  paving
  matroids have been conjecture to be ``most'' matroids
  in~\cite{mayhew+newman+welsh+whittle} while for the other classes for which Stanley's conjecture has been established, the proportion of
  matroids with $n$ elements in each class is negligible in comparison
  with the
  total number of matroids with $n$ elements as $n$ tends to
  infinity. Also, notice that any rank three simple matroid is paving.

  The problem of giving good lower bounds on the number of bases of a
  paving matroid appears to be a challenging but interesting problem.
  The function $f(r,n-r)$ gives
  us a lower bound for the number of bases of paving matroids, but is
  not tight in most cases. When $n-r=2$ and $n$ is even,
  $f(r,n-r)$ gives the lower bound of $n(n-2)/2$ for $n$ which is
  achieve by the  dual matroid of the 2-thickening of $U_{2,m}$; when $n$ is odd it
  gives the lower bound of $(n-1)^2/2$ which is
  achieved by the  dual matroid of the free extension of the
  2-thickening of $U_{2,m}$. So, in this case $f(r,n-r)$ is a tight
  lower  bound. But for the case of $n-r=3$ the situation is quite
  different. When rank is 2, the bound gives 6 bases, which are achieve
  by the paving matroid $U_{1,3}\oplus U_{1,2}$. When the rank is 3
  the lower bound gives 13 but there are 8 coloopless paving matroids
  with 6 elements and rank 3, yet the minimum number of bases is
  15. Even if we use $S(r,n)$ for obtaining a lower bound, we get 14
  in this case.

  The function $f(p,d)$  is very intriguing. The function itself seems
  very difficult to compute from the definition. We can prove that
  $f(p,d)=\overline{f}(p,d)$ for $d=1,2,3$ and all $p\geq 1$ and also
  for $d=4$ for $1\leq p\leq 6$.  We have checked that
  $\overline{f}(p,d)= L_2(p,d)$ for many small values of $p$ and $d$
  with $(p,d)>1$  by using  Maple.
  Conjecture~\ref{conjecture} would imply, for example that
  $f(p,d)=f(d,p)$ which
  geometrically is not so easy to see and we have been unable to prove.

\small

\setlength{\itemsep}{-.8mm}

redpajama_set_name: RedPajamaArXiv


text:
1/03/2016 - Press release
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A study published in the prestigious journal Scientific Reports from the Nature group demonstrates, for the first time and using computational tools, that polyunsaturated lipids can alter the binding rate of two types of receivers involved in certain nervous system diseases. The work was led by members of the Research Programme on Biomedical Informatics at the IMIM (Hospital del Mar Medical Research Institute) and Pompeu Fabra University as well as researchers from the University of Tampere (Finland), and also involved scientists from the University of Barcelona. Using latest-generation molecular simulations, which are like "computational microscopes", the researchers have demonstrated that a decrease in polyunsaturated lipids in neuronal membranes, as seen in Parkinson's and Alzheimer's sufferers, directly affects the binding rate of dopamine and adenosine receptors.
Més informació "Regulating neuronal membrane lipids could be the key to Alzheimer's and Parkinson's"
New genes formed continuously allowing acquisition of novel functions during evolution
Research just published in the journal Plos Genetics has found that gaining new genes during evolution is a much more frequent event than previously thought. It has been seen that there are hundreds of genes that might be unique to humans, and something similar occurs in chimpanzees. Some of these genes will be useful for the organism in question and the rest will disappear in time. The work was led by Mar Albà, an ICREA researcher at the Mar Institute of Medical Research (IMIM) and Jorge Ruiz-Orera, from the same group, together with researchers from Pompeu Fabra University (UPF) and the Centre for Genomic Regulation (CRG). For some time it was thought that all new genes originated from other genes, for example, from duplications of already existing genes. But recently it has been seen that some, the so-called de novo genes, originate in genomic regions that previously contained none. According to Mar Albà "This work shows that the formation of DNA motifs, through the accumulation of random mutations, would have been a determining factor in the emergence of new genes." DNA motifs are elements that activate gene expression.
Més informació "New genes formed continuously allowing acquisition of novel functions during evolution"
New computational approach to predicting adverse drug reactions with higher confidence
A new integrated computational method helps predicting adverse drug reaction—which are often lethal—more reliably than with traditional computing methods. This improved ability to foresee the possible adverse effects of drugs may entail saving many lives in the future. The study that is being conducted by researchers from IMIM (Hospital del Mar Medical Research Institute), Pompeu Fabra University, and the company Chemotargets, within the framework of the European eTOX project, was chosen for the cover of the journalChemical Research in Toxicology. Most computer tools employed today to detect possible adverse effects of compounds that are candidates for new medicines are based on detecting labile fragments in the drug's structure. These fragments can potentially transform to form reactive metabolites, which can have toxic properties. This is what is known as idiosyncratic toxicity and is a big headache for the pharmaceutical industry, as it tends to be detected in late development stages of the drug and even when it is already on the market, often causing the drug to be withdrawn.
Més informació "New computational approach to predicting adverse drug reactions with higher confidence"
De novo genes, how are they originated?
Some time ago it was believed that new genes originated, at least partially, from other genes; for instance, from the duplication of existing genes. But recently it has been seen that there are certain genes, those called de novo genes, that originate in genomic regions that didn't contain any genes previously. A study published in arXiv.org, led by Mar Albà, the coordinator of the Group on Evolutionary Genomics from the Research Programme on Biomedical Informatics (GRIB) from IMIM and the UPF and an ICREA Professor, and with the collaboration of scientists from different centres at the PRBB (Barcelona Biomedical Research Park), has revealed there are hundreds of de novo genes originated in humans, many of which where unknown until now. Some of these genes are coded through small proteins that are expressed in the brain or in germinal cells.
Més informació "De novo genes, how are they originated?"
Crucial step in AIDS virus maturation simulated for first time
Bioinformaticians at IMIM and UPF have used molecular simulation techniques to explain a specific step in the maturation of the HIV virions, essential in understanding how the virus replicates. These results, which have been published in the latest edition of PNAS, could be crucial to the design of future antiretrovirals.The work has been carried out using GPUGRID.net, a voluntary distributed computing platform leveraging GPU accelerators to deliver "virtual supercomputing" performance.
Més informació "Crucial step in AIDS virus maturation simulated for first time"
New public-private open innovation project
The project- based on a collaboration agreement between the Barcelona Science Park (PCB) and the companies SOM Biotech and Metasbio– will enable the discovery of new therapeutic applications of previously existing drugs (repurposing), especially in areas of great medical need and market potential, such as the central nervous system, oncology and orphan diseases.
Més informació "New public-private open innovation project"
GRIB participated at BIO Boston 2012, the top biotechnology convention in the world
"Connect, partner and innovate". This is the slogan for the twentieth edition of the BIO International Convention, held 18 to 21 June 2012 in the city of Boston (Massachusetts, United States). BIO is the most important international business meeting in the biotechnology arena. This year, it will bring together more than 15,000 delegates from 65 countries, more than 900 exhibitors and 1,800 companies. In addition to the exhibition area, it also features a Business Forum for partnering meetings, which in 2011 generated more than 21,000 one-to-one meetings.
Més informació "GRIB participated at BIO Boston 2012, the top biotechnology convention in the world"
INBIOMEDvision Training Challenge 2012
After one week of intensive training, the closing event of the INBIOMEDvision Training Challenge successfully took place in Barcelona. This training gathered together two teams of 5 PhD students with different nationalities and scientific backgrounds (chemists, medical doctors, informatics, mathematics, biologists etc.) to work on two case studies related with health sciences during one week at the Monastery of Les Avellanes (Lleida, Spain).
Més informació "INBIOMEDvision Training Challenge 2012"
GRIB EXPO: a unique opportunity to bring Biomedical Informatics closer to the clinical sector and to industrial innovation
Yesterday, Thursday 26 April, saw the conclusion in Barcelona of the first edition of GRIB EXPO, a symposium organised by the Research Programme on Biomedical Informatics (GRIB), a joint research group of the IMIM (Hospital del Mar Research Institute) and Pompeu Fabra University (UPF). The aim of the EXPO was to bring Biomedical Informatics closer to the clinical sector and to industrial innovation, with a view to promoting the open exchange of ideas and cooperation.
Més informació "GRIB EXPO: a unique opportunity to bring Biomedical Informatics closer to the clinical sector and to industrial innovation"
Identified 115 proteins that would allow designing new generation anti-cancer drugs
Researchers from the Research Programme on Biomedical Informatics (GRIB) from the IMIM and the Pompeu Fabra University (UPF) have identified 115 proteins via computer simulation that could be highly relevant to treat colon-rectal cancer, since they would make it possible to define the strategy to design new generation anti-cancer drugs. The study will be published in the journal PLoS ONE
Més informació "Identified 115 proteins that would allow designing new generation anti-cancer drugs"

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Freeing up of unmanaged code in C# I call a piece of an unmanaged C++ code from my C# application
to calculate fast fourier transform of a discrete time signal.
I make a call something like this
IntPtr ptr = ComputeFFTW(packetSig, packetSig.Length, (int)samplFrequency,(int)fftPoints);

    unsafe
     {
           double *dPtr = (double*)ptr;

           for(int l = 0; l < fftData.Length; l++)
        {
          fftData[l] = dPtr[l];
            }

      }

Though this snippet of code works fine and gives me the desired results, i can see that there is sort of performance hit (memory leak) is incurred while calculation is in progress. The CLR fails to reclaim the local (double) variables and my application gobbles up RAM space considerably.
Can anyone of you suggest the places where i might be doing it wrong.
From my side, I ran my application using ANTS Mem Profiler and i can see on the snapshot that the double objects nearly claim >150MB of the mem space. Is this a normal behaviour ??
Class Name  Live Size (bytes)   Live Instances
Double[]    150,994,980         3

Any help is appreciated in this regard
Srivatsa

A: Since the C++ function allocates memory you will have to manually free that chunk in your C# application (free the pointer). A better way to do invoke unmanaged code is to allocate all the variables and memory chunks (Temp parameters too) in your C# application and pass them to your C++ code as parameters. In this way you wont have any memory issues with your unmanaged code.

A: You can use Marshal.Copy(IntPtr, Double[], Int32, Int32) method to copy array of double values from unmanaged ptr to managed ffData array:
IntPtr ptr = ComputeFFTW(packetSig, packetSig.Length, (int)samplFrequency,(int)fftPoints);

Marshal.Copy(ptr, fftData, 0, fftData.Length);

If ComputeFFTW returns pointer to dynamically allocated memory, you need to release it after using. Make this in unmanaged code, add function like Release and pass ptr to it.

redpajama_set_name: RedPajamaStackExchange


College Station - convenient location.
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House for rent in College Station.
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MOVE IN READY Three BR, SPACIOUS HOME. Will Consider!
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text:
Q: Have Google maps center around geo-locs and zoom in appropriately I would like to pass an x amount of geo-locations to the Google Maps API and have it centered around these locations and set the appropriate zoom level so all locations are visible on the map. I.e. show all the markers that are currently on the map.
Is this possible with what the Google Maps API offers by default or do I need to resolve to build this myself?

A: I used fitBounds (API V3) for each point:

*

*Declare the variable.
var bounds = new google.maps.LatLngBounds();

*Go through each marker with FOR loop
for (i = 0; i < markers.length; i++) {
    var latlng = new google.maps.LatLng(markers[i].lat, markers[i].lng);
    bounds.extend(latlng);
}

*Finally call
map.fitBounds(bounds);

A: There are a number of ways of doing this but it is pretty easy if you are using the V2 API. See this post for an example, this group post or this post.

A: For V3 there's zoomToMarkers

A: Gray's Idea is great but does not work as is. I had to work out a hack for a zoomToMarkers API V3:
function zoomToMarkers(map, markers)
{
    if(markers[0]) // make sure at least one marker is there
    {
        // Get LatLng of the first marker
        var tempmark =markers[0].getPosition();

        // LatLngBounds needs two LatLng objects to be constructed
        var bounds = new google.maps.LatLngBounds(tempmark,tempmark);

        // loop thru all markers and extend the LatLngBounds object
        for (var i = 0; i < markers.length; i++)
        {
            bounds.extend(markers[i].getPosition());
        }

        // Set the map viewport
        map.fitBounds(bounds);
    }

}


Savage's Black Economy Backdrop Kit is the perfect setup for small studio space! Our black polyester background is sold individually or sold as an economical kit complete with a lightweight background stand and a complimentary carry bag. Backdrop is sized to 5ft x 9ft and made from a wrinkle-resistant matte material, offering a true black finish that doesn't gray out even under direct lights and high exposures.


CBOT futures closed steady to lower and close to the day's lows. Prices hit highs about 915 am CT and worked lower the rest of the trading day. Pressure came from ideas that improving weather will help corn and bean yields. The continued strengthening of the US Dollar is also negative for exports. Given the large and swift rally in CBOT corn/bean futures in recent weeks, it's not surprising to see the market pause and backfill some.
Cattle futures closed moderately lower on weak fundamentals as boxed beef values fall and fed cattle are expected to possibly trade down $2/cwt from last week's $150. Choice carcasses down 0.99 at 233.95; Select down 2.89 at 230.82. Plentiful pork and chicken supplies at lower prices are hurting the cattle/beef market.
Lean Hog futures closed mixed with little direction provided by the pork market. FOB Plant Pork closed up .10 at 82.65/cwt., an improvement from the midday quote.


The other main feature of your Chromebook desktop is the window system. When opened, almost all applications will load into a system of windows, much as they would on a Windows or Apple computer. The difference, though, is that the windows on a Chromebook are all Chrome web browser windows.
By default, a newly opened window appears maximized, which means that it fills the screen. You can close a window by clicking the X located in the top-right corner. Next to the X in the top-right corner of the window is a square icon.
Clicking the leftmost button shrinks the window so that it occupies only the left half of the screen.
Clicking the middle button minimizes the window, meaning it will remove the window from the screen, hiding it behind the Chrome icon on your shelf. To restore the window, just click the Chrome icon.
Clicking the rightmost button shrinks the window so that it occupies only the right half of the screen.


text:
Belfast City Council
Who is Alliance councillor Kate Nicholl, the new Belfast Lord Mayor?
The 33-year-old succeeds DUP councillor Frank McCoubrey as the city's first citizen
Brendan Hughes
Alliance Party councillor Kate Nicholl
Alliance Party councillor Kate Nicholl is the new Lord Mayor of Belfast.
The 33-year-old succeeds DUP councillor Frank McCoubrey as the city's first citizen.
DUP councillor Tom Haire is the new deputy lord mayor, succeeding the SDLP's Paul McCusker.
Ms Nicholl, who was born in Zimbabwe, is the first person in recent times to hold the mayoral position who was not born in the UK or Ireland.
The councillor for the Balmoral area of South Belfast has taken a career break from her job in marketing to fulfil her year-long tenure as mayor.
Ms Nicholl is married and has a one-year-old son, Cian.
The new mayor's mother is from South Africa and her father from Co Down in Northern Ireland.
She grew up in Marondera in Zimbabwe, but moved with her family to Belfast at the age of 12 when violence broke out in Zimbabwe in 2000.
Belfast council is about more than just bins and dog fouling, says new Lord Mayor
Hillsborough becomes first place in NI to receive Royal status
Ms Nicholl attended Methodist College in Belfast, and has worked as a researcher for former Alliance MLA Anna Lo.
She was originally co-opted to the council, but in the 2019 local elections she topped the poll in her area.
Ms Nicholl will also be an Alliance candidate for the South Belfast constituency in next year's Assembly election.
The Alliance councillor has described becoming Lord Mayor as a "positive message to all other newcomers".
She said: "Belfast is a diverse, vibrant multi-cultural city and I think the fact that the lord mayor of Belfast, the first citizen, is going to be someone who wasn't born here, who has chosen to make it home, and who has really embraced the city, is really positive."
The lord mayor acts as the chairperson of Belfast City Council and represents the local authority at civic events.
The position is rotated annually between the parties represented on the council based on their number of seats.
Ms Nicholl has said her theme for her year as Lord Mayor will be "Our Belfast".
She hopes to highlight the services the council provides.
Ms Nicholl also plans to have a "monthly mayor's environmental challenge" set in conjunction with organisations which focus on environmental work across the city.
And the new mayor intends to profile and volunteer with charities on a regular basis over the coming year.
Mayoral portrait of Sinn Fein minister to cost ratepayers up to £13,000
All NI council decisions on rates for 2021/22 - what it means for you


I realise it maybe wasn't quite right for you guys but I think it sounds like something I would enjoy, so thanks for sharing!
On the surface, this sounds like it should have been a more pleasing reading experience. Sorry this was more of a miss.
I love the cover, and I love your reviews ladies! I know for sure this would be over my head. A challenge for sure!
Thank you, T! Yeah, Probably not the best one for a casual reader.
This one has been on several lists, which is what brought it to my attention. I do hope to get to it at some point, but I'm glad I read your reviews first so I know it's going to be challenging from the onset.
I think for sure it will appeal to a large audience! I hope you enjoy it when you get to it!!
Thank you, Lindsay! I wish you had too!


text:
Comey compares Trump to mob boss, Trump cries…
Comey compares Trump to mob boss, Trump cries "slime ball"
In this June 8, 2017 file photo, former FBI director James Comey speaks during a Senate Intelligence Committee hearing on Capitol Hill, in Washington. Comey is blasting President Donald Trump as "unethical and untethered to truth," and says Trump's leadership of the country is "ego driven and about personal loyalty." Comey's comments come in a new book in which he casts Trump as a mafia boss-like figure who sought to blur the line between law enforcement and politics. (AP Photo/Andrew Harnik, File)
April 13, 2018 at 7:59 a.m.
By CHAD DAY and JONATHAN LEMIRE
WASHINGTON — Firing back at a sharply critical book by former FBI director James Comey, President Donald Trump blasted him Friday as an "untruthful slime ball," saying, "It was my great honor to fire James Comey!"
Trump reacted on Twitter early Friday, the day after the emergence of details from Comey's memoir, which says Trump is "untethered to truth," and describes him as fixated in the early days of his presidency on having the FBI debunk salacious rumors he said were untrue but that could distress his wife.
The book, "A Higher Loyalty," is to be released next week. The Associated Press purchased a copy this week.
In the book, Comey compares Trump to a mafia don and calls his leadership of the country "ego driven and about personal loyalty."
Comey also reveals new details about his interactions with Trump and his own decision-making in handling the Hillary Clinton email investigation before the 2016 election. He casts Trump as a mobster-like figure who sought to blur the line between law enforcement and politics and tried to pressure him personally regarding his investigation into Russian election interference.
The book adheres closely to Comey's public testimony and written statements about his contacts with Trump and his growing concern about Trump's integrity. It also includes strikingly personal jabs at Trump that appear sure to irritate the president.
The 6-foot-8 Comey describes Trump as shorter than he expected with a "too long" tie and "bright white half-moons" under his eyes that he suggests came from tanning goggles. He also says he made a conscious effort to check the president's hand size, saying it was "smaller than mine but did not seem unusually so."
"Donald Trump's presidency threatens much of what is good in this nation," Comey writes, calling the administration a "forest fire" that can't be contained by ethical leaders within the government.
On a more-personal level, Comey describes Trump repeatedly asking him to consider investigating an allegation involving Trump and Russian prostitutes urinating on a bed in a Moscow hotel, in order to prove it was a lie. Trump has strongly denied the allegation, and Comey says that it appeared the president wanted it investigated to reassure his wife, Melania Trump.
Trump fired Comey in May 2017, setting off a scramble at the Justice Department that led to the appointment of Robert Mueller as special counsel overseeing the Russia investigation. Mueller's probe has expanded to include whether Trump obstructed justice by firing Comey, which the president denies.
Trump has assailed Comey as a "showboat" and a "liar." Top White House aides also criticized the fired FBI director on Friday. White House spokeswoman Sarah Huckabee Sanders questioned Comey's credibility in a tweet and White House counselor Kellyanne Conway said Comey took "unnecessary, immature pot shots."
Comey's account lands at a particularly sensitive moment for Trump and the White House. Officials there describe the president as enraged over a recent FBI raid of his personal lawyer's home and office, raising the prospect that he could fire Deputy Attorney General Rod Rosenstein, who appointed Mueller, or try to shut down the probe on his own. The Republican National Committee is poised to lead the pushback effort against Comey by launching a website and supplying surrogates with talking points that question his credibility.
Trump has said he fired Comey because of his handling of the FBI's investigation into Clinton's email practices. Trump used the investigation as a cudgel in the campaign and repeatedly said Clinton should be jailed for using a personal email system while serving as secretary of state. Democrats, on the other hand, have accused Comey of politicizing the investigation, and Clinton herself has said it hurt her election prospects.
Comey writes that he regrets his approach and some of the wording he used in his July 2016 press conference in which he announced the decision not to prosecute Clinton. But he says he believes he did the right thing by going before the cameras and making his statement, noting that the Justice Department had done so in other high profile cases.
Every person on the investigative team, Comey writes, found that there was no prosecutable case against Clinton and that the FBI didn't find that she lied under its questioning.
He also reveals new details about how the government had unverified classified information that he believes could have been used to cast doubt on Attorney General Loretta Lynch's independence in the Clinton probe. While Comey does not outline the details of the information — and says he didn't see indications of Lynch inappropriately influencing the investigation — he says it worried him that the material could be used to attack the integrity of the probe and the FBI's independence.
Comey's book will be heavily scrutinized by the president's legal team looking for any inconsistencies between it and his public testimony, under oath, before Congress. They will be looking to impeach Comey's credibility as a key witness in Mueller's obstruction investigation, which the president has cast as a political motivated witch hunt.
The former FBI director provides new details of his firing. He writes that then-Homeland Security Secretary John Kelly — now Trump's chief of staff — offered to quit out of disgust at how Comey was dismissed. Kelly has been increasingly marginalized in the White House and the president has mused to confidants about firing him.
Comey also writes extensively about his first meeting with Trump after the election, a briefing in January 2017 at Trump Tower in New York City. Others in the meeting included Vice President Mike Pence, Trump's first chief of staff, Reince Priebus, Michael Flynn, who would become national security adviser, and incoming press secretary, Sean Spicer. Comey was also joined by NSA Director Mike Rogers, CIA Director John Brennan and Director of National Intelligence James Clapper.
After Clapper briefed the team on the intelligence community's findings of Russian election interference, Comey writes, he was taken aback by what the Trump team didn't ask.
"They were about to lead a country that had been attacked by a foreign adversary, yet they had no questions about what the future Russian threat might be," Comey writes. Instead, he writes, they launched into a strategy session about how to "spin what we'd just told them" for the public.
Comey says he had flashbacks to his time investigating the Italian Mafia as a federal prosecutor in Manhattan, thinking that Trump "was trying to make us all part of the same family."
"For my entire career, intelligence was a thing of mine and political spin a thing of yours. Team Trump wanted to change that," he writes.


Ticket entitles the student late entry into the event.
Early Bird Competition entry is open until 12:00am on Saturday 22nd June.
Late Entires are available from 12:00am on Saturday 22nd June to 12:00am on Friday 28th June.
Registration will close after this time.
PLEASE NOTE, you must enter the competitors name, not the name of the person completing the registration i.e. the parent. Division are arranged based on rank, age & height. Pleae enter accuate details as incorrect information may lead to disqualification.
Competitors aged 13 & 14 may be put into the Juniors Divisions based on their height.
Competitors aged 11 & 12 and 16 & 17 may be put into the Teengers Divisions based on their height.
In any of the above cases, you will be contacted.
All Competitors are encouraged to arrive at least 15 minutes before their Group is due to begin. Although Instructors will do their best to make sure divisions run as scheduled, there may be minor time delays depending on the amount of competitors in each division. Orange to Black Belts (8th KUP & above) will have Forms (Patterns), Point Sparring & Board Breaking Competitions within their divisions. White & Yellow Belts (10th & 9th KUP) will have Forms & Step Sparring (One Steps) in their divisions. Teenager & Adult White & Yellow Belts may enter the Sparring and/or Board Breaking, however must select the Orange to Black Belt Ticket when registering.
I have full knowledge that by nature tournament participation can frequently include strenuous exercise and body contact along with associated dangers. In recognition of the possible dangers connected with any kind of physical activity, member(s) hereby knowingly and voluntary waive any right of cause or action of any kind whatsoever arising as a result of such activity from which any liability may or could accrue to the Basingstoke school of martial arts, Phoenix School of Martial Arts (SMA) or Reading Taekwondo Academy, their officers, agents, employees or instructors or any other club/school that is taking part in this event.


Familiarize yourself with the 2019 BMW 330i wagon This 4 door, 5 passenger sedan leads among competitors in its segment. Performance and efficiency are both prioritized thanks to the 2 liter 4 cylinder engine, providing a smooth and predictable driving experience. A turbocharger is also included as an economical means of increasing performance. A wealth of standard features means that you no longer have to sacrifice. Like heated seats, a tachometer, speed sensitive wipers, an automatic dimming rear-view mirror, power front seats, fully automatic headlights, lane departure warning, and a blind spot monitoring system. For drivers who enjoy the natural environment, a power moon roof allows an infusion of fresh air. BMW ensures the safety and security of its passengers with equipment such as: head curtain airbags, front side impact airbags, traction control, brake assist, a panic alarm, an emergency communication system, and 4 wheel disc brakes with ABS. For added security, dynamic Stability Control supplements the drivetrain. Our knowledgeable sales staff is available to answer any questions that you might have. Please don't hesitate to give us a call.


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Much of modernity and even more of post-modernity is energized by the confrontation with the metaphysical abyss. Poetry has responded in myriad ways, ranging from a retreat into subjectivity to fragmented or intentionally nonsensical poetry that mirrors or enacts a world perceived as fragmented and nonsensical. In Accepting the Disaster, Joshua Mehigan adopts a different approach. He writes extremely controlled poems whose discipline and understatement belie the depths of passion that inform them. He accepts the disaster; he describes the disaster. He does not join in it.
to some failing of my own.
but nothing, ever, anymore, for you.
Mehigan's world is harsh and uncomfortable, but it is also ours, stripped of cherished metaphysical consolations. Mehigan's poetry expresses a common and rich worldview, and in Accepting the Disaster, he has done so elegantly, thoroughly, relentlessly, profoundly and beautifully.
Accepting the Disaster is not a narrative, but it tells the story of post-industrial decay in a small city undoubtedly based on the town in upstate New York where Mehigan grew up. The opening poem, Here, establishes the setting as a place where "nothing . . . ever changes, till it does." Although the lives of the townspeople seem both mirthless and pointless, change is not progress, unless it is the sort of progress that Walter Benjamin has evoked in describing a figure in a drawing by Paul Klee: "His face is turned toward the past. Where we perceive a chain of events, he sees one single catastrophe which keeps piling wreckage upon wreckage and hurls it in front of his feet. The angel would like to stay, awaken the dead, and make whole what has been smashed. But a storm is blowing from Paradise; it has got caught in his wings with such violence that the angel can no longer close them. The storm irresistibly propels him into the future to which his back is turned, while the pile of debris before him grows skyward. This storm is what we call progress."
Mehigan's book alternates between poignantly detailing the disaster of a post-industrial town and hinting at a broader cultural disaster. When he writes in the latter mode, as in the title poem, he writes in conversation with Benjamin. Because Mehigan recognizes the dark humor and absurdity in our experience of the disaster, Benjamin is Mehigan's straight man.
half the world ranked above world averages.
Elsewhere, the disaster consists of very real but familiar concerns about mortgages, past-due notices, and sleep aids, or it lapses into surrealistic absurdity, as in "we spent our dotages / becoming ostriches and walruses." But the poem's real achievement is the elision of imagined catastrophe and everyday life.
Seas rose, and threatened to deluge our lowlands and isthmuses.
Our western forests turned to furnaces.
Equinoctial dust stopped up our sinuses.
and, each month, one or two more languages.
Mehigan's poem defamiliarizes his generation's familiar disaster, but now and then he refamiliarizes and thus reshapes it. As Mehigan presents it, one generation experiences the disaster and reminisces about it in conversations with mocking children. But the disaster is also with us still, and while the signs of disaster pervade our daily lives, we are unprepared to do anything more than accept it.
Other poems detail the town's more local disaster. "The Smokestack" accretes meaning to the smokestack, which together with its church spire is one of the town's main landmarks, through a litany of metaphors that are by turns arresting and comic in their extravagance and diversity. The smokestack may belong to another fixture of the town, commemorated in "The Cement Plant," which provides an example of what change means in a town like this. The cement plant sustains the men of the town, while killing some of them and coating the rest and the town itself with its dust. In "Work Song," we get a succinct description of the town's culture of work: "We come. We carp. We're fired. We worry later." In "The Dream Job," Mehigan illustrates how the monotony and tedium of work in the town bleeds over into the monotony of home life and even into dreams. A town where nothing ever changes might, despite its tedium, have its familiar comforts and consolations. But the cement plant closed down, leaving the workers with their worry, their dust, their bodies and relationships wrecked by their lives' merciless drudgery, and very little else. They have no real prospects for future gainful employment or even pleasurable ways to pass the time.
However else the town might have been experienced by its other inhabitants, the town that Mehigan describes is the one in which he grew up.
Many of Mehigan's poems in Accepting the Disaster describe places, and such descriptions serve as an introduction to the townspeople. As Mehigan puts it in his eerie and wise "The Sponge," "minds and their milieus are all one thing." Other poems, "The Crossroads" and "Down in the Valley," describe mini-catastrophes, a car accident and (one supposes) a sexual assault, within the town's more general disaster. The places resonate with the horrific events that they hosted.
The book concludes with a lyrical poem, "Shark's Tooth," which is written in a different key from the rest. The speaker in the poem has found a shark's tooth on a sandy beach. The shark's tooth seems like a prize, like something the speaker searched for and found. He attempts to re-create the conditions that led him to the shark tooth so that he can find another, but at the same time he holds on to his shark tooth "like a sort of charm, / a talisman to ward off superstition." But the speaker's attachment to the shark tooth is itself a superstition. He contemplates the object as it sits in his palm and asks, "Was it an omen?" The poem never resolves the mystery of the shark tooth's allure, concluding that it is "like a gift, or like bait," held out to the speaker. The poem, with its lush descriptions of beachfront serenity and inscrutable skies, seems, for Mehigan, uncharacteristically hopeful. But as the last line suggests, Mehigan is ambivalent about hope.
Mehigan's second book was ten years in the making. Each poem evidences his craftsmanship. There are no false notes; no mistakes in word choice; no metrical infelicities. Mehigan is a master artist working at the height of his power. Fittingly, given Mehigan's dour Weltanschauung, he realizes his gifts in an era when the world, even the world of poetry, is largely indifferent to the type of art that he has mastered. Already reviewers are predicting that his poems will make their way into anthologies and so on to posterity. I hope that they are right. But while Mehigan's admirers agree that Mehigan's poems are irreproachable, many readers will find them unapproachable. Mehigan hides his genius behind plain language and unpretentious subject matters. Mehigan writes with such facility in traditional forms that his poems do not call attention to themselves as adhering to a tradition. His mastery of form is so advanced that he can often disguise even the most familiar forms and compose in strict meters that will escape the notice of the casual reader, who will as a result be deaf to Mehigan's music.
Because there are so few contemporary readers of poetry, and because those few readers often have not been taught to appreciate formal poetry (and are sometimes taught that formal poetry is antiquated or conservative), Mehigan writes for a small, if highly appreciative audience. His poems will not attract readers for their subject matter, for their explorations of the poet's interiority, ethnicity, sexuality, hardships, addictions, romances or break-ups. Nor do Mehigan's poems have a clear political valence. If a reader is not impressed by Mehigan's formal virtuosity, she might be put off by his poems' austerity, their insistent gloom, and their resignation in the face of collapse. Mehigan's poems will only be read because nothing better will be written, ever, anymore, for us. Let us hope that is reason enough.
D.A. Jeremy Telman is associate dean for faculty development and professor of law at the Valparaiso University Law School and the editor of theContractsProf Blog, to which he contributes, among other things, limericks that summarize cases that he covers in his courses on contracts and business organizations. His scholarly writings have appeared in Austrian, French, German, Indian, Israeli, South African and U.S. publications.


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State Considering Centralia's Lakeview Inn to Host COVID-19 Isolation,
Posted on May 1, 2021 by ericadminchia
By Eric Rosane / ericr@chronline.com
The state Department of Health says it's finalizing talks with a Centralia inn to open an isolation and quarantine facility for travelers who contract COVID-19 while in transit through Washington state.
The transition is expected to start the first week of May at Lakeview Inn, said Wayne Clifford, medical surge branch director with the Department of Health. A contract is still being negotiated, with plans to lease the entire 40-bed hotel.
The facility would also serve shipping and vessel crews, state work release inmates and provide occasional support to local municipalities.
Camp Solomon Schechter has hosted the Department of Health's program for a few months now, Clifford said, but has recently voiced plans to reopen for the season. That led the state health department to look at other options in the area that were close to SeaTac airport.
"Hotels really provide an optimal structure for us with all the rooms and the ability to keep people apart and take care of them, and they have all the (utilities) they need inside," Clifford told the Centralia City Council during a meeting Tuesday.
The program is completely voluntary for those workers and travelers in transit through the state who contract the viral disease. People usually stay between 10 to 14 days in the facilities, and Department of Health staff are at the site at all times.
The contract could last through the end of this year, Clifford said, with plans to look at the Maple Lane campus as a long-term solution.
"I think, over the course of that time, we'll certainly understand if we're going to continue to need that capability, and if it's working out or not working out. We're going to work closely with the city and the county as we move forward with this to make sure it's working for everyone," he said.
Shamsher Singh, owner of the Lakeview Inn in Centralia, said over the phone Friday that talks are still fairly preliminary between him and the Department of Health.
While they're still waiting on final contracts, Singh said they've been busy cleaning the facility over the last couple days in anticipation for the Department of Health's arrival. Gates currently line the perimeter of the property, and the parking lot is vacant.
"Up to now, by noon time, there's no done deal yet … I haven't gotten any paperwork or any phone calls from them yet," Singh said. "I'm here for the (Department of) Health, doing whatever I can in this tough time."
Singh said they have not come to an agreement yet on the leasing price of the facility, but noted he expects they will finalize the contract within the next week.
Lakeview Inn's rooms will be outfitted and accustomed to meet the Department of Health's specifications.
Read More: State Considering Centralia's Lakeview Inn to Host COVID-19 Isolation,
Posted in NewsTagged Centralias, COVID19, Host, Inn, Isolation, Lakeview, state
As Employees Return, Counter Work-From-Home FatigueCrowded conditions for migrant children at border facilities


The following report analyses the overall strategic management procedure followed by the Japan based world-renowned car manufacturing company Toyota Motor Corporation. The company is largely valued due the unique production system implemented by it in providing high quality satisfaction to its global customers. However, Toyota faced difficulties in manufacturing different type of car models belonging to the same production line.
The California Department of Motor Vehicle is a state owned agency responsible for issuing licenses to the vehicles and vessels in California. It also regulates and monitors the driving schools, traffic schools and the personal vehicle owners. The agency also performs the task of vehicle registration, renewable of licenses, collection of registration fees and formulating rules and regulations with regards to vehicles and their usage. The agency is a state owned body, so it had a monopoly in the country. But they want to provide efficient services to their customers. For this reason, the director of the company along with the steering committee took steps to plan a business re-engineering project. They monitor each aspect of the business functions of the agency thoroughly. This is done to find those weak points which required to be changed. In this situation when the agency's motive is to increase their quality of service and performance, re-engineering of the business process is done. Re-engineering involves uprooting the weak points in the system totally, planning out new framework for the solution and finally implementing the designed solution. The business policies supporting these systems also need to be changed along with the new framework. This will not only bring productivity but enhance the performance level of the company as a whole.
The following report analyses the strategic group of the UK based world-renowned retailer Marks & Spencer. The company has gained widespread popularity among its global customers with its home furnishing and clothing division and the food segment. The primary philosophy of the company emphasises on providing high quality service satisfaction to its customers.


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Great Leaders of India- their Contribution can't be explained in Words!
Some Great Leaders of India – Ever Indian Will remember them forever!
India is the land of great leaders, where many scientists and many freedom fighters have been born and died, whose contributions cannot be explained, who led us on the path of development, the path to success, and who ruled the country very well. We are literally very proud to have such great leaders. Each and every leader had played a role in changing the world respectively. Here I am mentioning some of the great leaders which cannot be forgotten and are highly praised by the country.
Mahatma Gandhi – Great Leaders of India
Mahatma Gandhi- every small kid knows about Mahatma Gandhi, the Father of Nation. Mohandas Karamchand Gandhi was born on 2nd October 1869 in Porbandar and was a lawyer, politician, and anti-colonial nationalist. Gandhi Ji was married at the age of 13 to Kasturba. Gandhi Ji studied Law in London and then went to South Africa for practice, he got to know that Indian citizens were denied their basic human rights and were discriminated against. When Gandhi Ji came back to India, he led various movements like Stayagrah Andolan, Khilafat movement, Non- Cooperation movements, Civil-Disobedience Movement and Quit India Movement. Gandhi Ji helped in India's independence from British rule through non-violent or peaceful movements.
Jawaharlal Nehru – Great Leaders of India
Jawaharlal Nehru- Jawaharlal Nehru was born on 14 November 1889, one of the greatest leaders who contributed a lot even their entire life. He was not just only a freedom fighter but was also a great politician and a great leader. Nehru Ji was the first Prime Minister of the country and served India till his death in 1964. He was also called Chacha Nehru because he loves children and said, that children are the real assets and strength of the country. He provided compulsory free primary child education in the five-year plan. Nehru Ji was also known as Pandit Nehru due to his connections to the Pandit community of Kashmir.
Lal Bahadur Shastri
Lal Bahadur Shastri Great Leaders of India
Lal Bahadur Shastri- Shastri Ji was born on 2nd October 1904, Mughal Sarai. Shastri Ji was the second Prime Minister of India, he led the White movement whose main aim was to increase the supplement of Milk by helping the Amul Milk, additionally, he also encouraged the green movement Revolution in India in 1965 to increase the food supplements. Shastri Ji was also a freedom fighter and participated in various movements led by Mahatma Gandhi Ji.
Subhash Chandra Bose
Subhash Chandra Bose Great Leaders of India
Subhash Chandra Bose- Subhash Chandra Bose was born on 23 January 1897, in Cuttack. He was also known as Netaji Bose. He was known for his big contributions to India's Independence by forming the Azad Hind Fauj or the National Army to fight the Britishers. He died on 18 august 1945, before the Independence in Taipei, Taiwan due to a third-degree burn in a crash.
Lala Lajpat Rai great leaders of India
Lala Lajpat Rai- Lala Ji was born on 28 January 1865, he was also known by the Dhudike. Lala Ji played a major role in India's Independence movements. He also led the Non-cooperation movement and Punjab Protest against the Jallia Vala Bagh incident. He also helped in the education and establishment of schools in the country. He died in 1928 in the brutal lathi charge by the Britishers on a peaceful protest against the Simon commission.
There were also many leaders who contributed to India's freedom from the Britishers and fought them till the end of their lives such as Rani Laxmi Bai, Mangal Pandey, Bhagat Singh, Chandra Shekhar Azad, Bal Gangadhar Tilak, Rabindra Nath Tagore, and Bankim Chandra Chatterjee and many others who had a major contribution in India's freedom.
APJ Abdul Kalam
APJ Abdul Kalam Great Leaders of India
APJ Abdul Kalam- Dr. Kalam was born on 15 October 1931. He was an aerospace engineer or rocket Scientist. Dr. Kalam served ISRO, Indian Space and Research Organization, and DRDO, Defense Research and Development Organization for many years. Dr. Kalam was a scientist, researcher, politician, and author who inspired millions of people. Kalam sir was also known as the Missile Man due to his major discoveries and inventions that helped in the development of the country through the development of India's Missile and Nuclear weapons. Later he served India as the 11th President of the country.
Indira Gandhi great leaders of India
Indira Gandhi– India Gandhi was born on 19 November 1917, and became the third and first women Prime Minister of the country after the Independence and till the date. Indira Gandhi was the daughter of the first Prime Minister Jawaharlal Nehru. She was awarded the Bharat Ratna for the victory of India against Pakistan. She served the Indian Nation for so long after her father Jawaharlal Nehru. She brought India on the path to becoming a nuclear power with the first Nuclear power test.
Sarojini Naidu Great leaders of India
Sarojini Naidu- Naidu was born in 1879 and was an Indian Politician, Patriot, writer, and administrator. She supported women for their civil rights, later Naidu joined the Indian Nationalist movement and Gandhi's Satyagraha Movement against the Britishers. You know, 13 February is the Indian National Women's day in India to respeSct Naidu's contribution to the Indian women. Later, she became the first Indian women President of the country.
Mother Teresa – Great Leaders of India
Mother Teresa– Teresa was born on 26 August 1910 in Skopje then she moved to India and lived the rest of her life. We already know why Mother Teresa is famous because of her major contributions to society. Mother Teresa founded the Missionaries of Charity to help poor people and children and provide free services to poor people. She also founded the Nirmala Shishu Bhavan, the first children's home to take care of homeless children. Teresa was known by the world for her social contributions to the country as well as to the world. Later she was awarded the Padma Shri, Bharat Ratna, and also won the Nobel Peace Prize in 1979. She is called the greatest humanitarian of the 20th century.
Bhimrao Ramji Ambedkar
Bhimrao Ramji Ambedkar- was born on 14 April 1891 and was a jurist, social reformer, economist, and politician. He fought against the social discrimination towards the Dalits who were considered untouchable, and Ambedkar was strongly against the caste systems. Ambedkar was the first minister of Law and justice in the country after Independence and the chief architect of the constitution of India.
I have already mentioned a few of the national personalities who have contributed to the country and it is been a lot. But apart from above all there are other leaders of the country who did their part in the development of the country and raise it from the small. Without these leaders, I do not think that anything would have been possible for us if those leaders weren't there for us. Now we are independent and living our lives just because our country got freedom from British rule and our freedom fighters fought till their death.
Tags:APJ Abdul KalamBal Gangadhar TilakBankim Chandra ChattarjeeBhagat SinghBhimrao Ramji AmbedkarChandra Shekhar AzadIndira GandhiJawaharlal NehruLal Bahadur ShastriLala Lajpat RaiMahatma GandhiMangal PandeyMother TeresaRabindra Nath TagoreRani Laxmi BaiSarojini NaiduSubhash Chandra Bose
Vedhika Latest Beautiful Instagram photos!
Top 10 Best Rappers in India in 2023!


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Write once read many (WORM) is een type gegevensdrager dat men één keer beschrijft, en dat vervolgens vele malen gelezen kan worden.

WORM is inherent aan de technische werking van sommige gegevensdragers, waarbij het medium feitelijk niet in staat is herbeschreven te worden. Ponskaarten zijn bijvoorbeeld WORM omdat je de geponste gaatjes niet kunt herstellen. cd-r's en dvd-r's zijn ook maar één keer te beschrijven. In de fabriek geperste compact discs en dvd's zijn geen WORM aangezien ze in het geheel niet door een computer zijn te beschrijven.

Sommige gegevensdragers kunnen wel meerdere keren beschreven worden, maar dat is in sommige situaties onwenselijk, zoals voor het maken van back-ups en archieven. Om ongewild wissen te voorkomen is de mogelijkheid tot WORM toegevoegd aan meervoudig beschrijfbare media als magneetbanden en harde schijven.

Opslagmedium

redpajama_set_name: RedPajamaWikipedia


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Welcome to the Village of Hoffman, Illinois!
techknow2023-01-10T21:39:55+00:00
Village President
Wm A. Guile – bguile@frontiernet.net
On behalf of the Village of Hoffman, I welcome you to our new website and the Village of Hoffman. As this website is developed village residents will have access to village information, events, ordinances, and chairpersons of various committees on the village board, and on line payment of water bills in the near future.
It is also the village board's intent to allow various organizations to notify the village and request information to be posted on the calendar of events page that will directly affect the residents of the village. Some examples are the town wide rummage sale – the Hoffman Fire Protection District fundraising events, HYSA events and schedules.
Within our website, residents and visitors can find information on our religious community and the various education and local business available to the Village of Hoffman residents and visitors to our community.
Ordinance for Operation of Non Highway Vehicle(s)
Schedule of Meetings - May 2022 to April 2023
Village of Hoffman 2020 Consumer Confidence Report
Residential Bulk Items Restrictions


Sharemoney is the international money transfer service of Omnex Group, Inc. helping people save money on remittances. Sharemoney is currently sending money to Philippines, Guatemala, Brazil, Vietnam and Dominican Republic. We have one of the most extensive and reliable networks of transfer agents and payers in the industry: approximately 1,000 locations in the U.S. and Canada, and points of payment in North, Central and South America, the Caribbean, Europe, Africa, Asia and the Middle East.
Omnex Group, who has been in the money transfer business for over 20 years, are a licensed money transmitter who send over a billion dollars each year in money remittances.