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%\newcommand{\pseudoconcave}{concave\xspace}
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\begin{document}

\begin{center}
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\begin{center}
\vskip 1cm{\LARGE\bf Enumeration of Concave Integer Partitions}
\vskip 1cm
\large
Jan Snellman and Michael Paulsen\\
Department of Mathematics\\
Stockholm University\\
SE-10691 Stockholm, Sweden \\
\href{Jan.Snellman@math.su.se}{\tt Jan.Snellman@math.su.se}
\end{center}

\vskip .3 in
\centerline{\bf Abstract}

\begin{quote}
  An integer partition \(\PartitionOf{\lambda}{n}\) corresponds, via
  its Ferrers diagram, to an artinian monomial ideal \(I \subset
  \C[x,y]\) with \(\dim_\C \C[x,y]/I = n\). If \(\lambda\) corresponds
  to an integrally closed ideal we call it \emph{\concave}. We study
  generating functions for the number of \concave partitions,
  unrestricted or with at most \(r\) parts.
\end{quote}

\vskip .3in

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% \begin{document}

% \title{Enumeration of concave integer partitions}
% \author{Jan Snellman}
% \address{Department of Mathematics\\
% Stockholm University\\
% SE-10691 Stockholm, Sweden}
% \email{Jan.Snellman@math.su.se}
% \author{Michael Paulsen}
% \begin{abstract}
%   An integer partition \(\PartitionOf{\lambda}{n}\) corresponds, via
%   its Ferrers diagram, to an artinian monomial ideal \(I \subset
%   \C[x,y]\) with \(\dim_\C \C[x,y]/I = n\). If \(\lambda\) corresponds
%   to an integrally closed ideal we call it \emph{\concave}. We study
%   generating functions for the number of \concave partitions,
%   unrestricted or with at most \(r\) parts.
% \end{abstract}
% 
% 
% \maketitle
% 


\begin{section}{\concave partitions}
  By an \emph{integer partition} \(\lambda = (\lambda_1, \lambda_2,
  \lambda_3,\dots)\)
  we mean a weakly decreasing sequence of non-negative integers, all
  but finitely many of which are zero. The non-zero elements are
  called the \emph{parts} of the partition. When writing a partition,
  we often will only write the parts; thus \((2,1,1,0,0,0,\dots)\) may
  be written as \((2,1,1)\).

  We write
  \(r=\NumParts{\lambda}\)
  for the number of parts of \(\lambda\), and \(n=\w{\lambda}=\sum_{i}
  \lambda_i\); equivalently, we write \(\PartitionOf{\lambda}{n}\)
  if \(n=\w{\lambda}\). The set of all partitions is denoted by
  \(\Partitions\), and the set of partitions of \(n\) by
  \(\Partitions(n)\). We put
  \(\card{\Partitions(n)}=p(n)\). By subscripting any of the above
  with \(r\) we
  restrict to partitions with at most \(r\) parts.

  We will use the fact that \(\Partitions\) forms a monoid
  under component-wise addition.

  For an integer partition \(\PartitionOf{\lambda}{n}\) we define its
  \emph{Ferrer's diagram} \(F(\lambda) = \setsuchas{(i,j) \in \Nat^2}{i
    &lt; \lambda_{j+1}}\). In figure~\ref{fig:exejkonk} the black dots
  comprise the Ferrer's diagram of the partition \(\mu=(4,4,2,2)\).


Then \(F(\lambda)\) is a finite \emph{order ideal}
  in the partially ordered set \((\Nat^2,\le)\), where \((a,b) \le
  (c,d)\) iff \(a \le c\) and \(b \le d\). In fact, integer partitions
  correspond precisely to finite order ideals in this poset.

  The complement \(I(\lambda)=\Nat^2 \setminus F(\lambda)\) is a
  monoid ideal in the additive monoid \(\Nat^2\). Recall that for a
  monoid ideal \(I\) the \emph{integral closure} \(\bar{I}\) is
  \begin{equation}
    \label{eq:intcl}
    \setsuchas{\vek{a}}{ \ell \vek{a} \in I \text{ for some } \ell &gt; 0}
  \end{equation}
  and that \(I\)
  is \emph{integrally closed} iff it is equal to its integral closure.

  \begin{definition}\label{def:ic}
    The integer partition \(\lambda\) is \emph{\concave} iff
    \(I(\lambda)\) is integrally closed. We denote by
    \(\bar{\lambda}\) the unique partition such that
    \(I(\bar{\lambda})= \overline{I(\lambda)}\).
  \end{definition}

  Now let \(R\) be the complex monoid ring of \(\Nat^2\).
  We identify \(\Nat^2\) with the set of commutative monomials in the
  variables \(x,y\), so that
  \(R\simeq \C[x,y]\). Then a monoid ideal \(I \subset \Nat^2\)
  corresponds to the monomial ideal \(J\) in \(R\) generated by the monomials
  \(\setsuchas{x^iy^j}{(i,j) \in I}\). Furthermore, since the monoid
  ideals of the form \(I(\lambda)\) are precisely those with finite
  complement to \(\Nat^2\), those monoid ideals will correspond to
  monomial ideals \(J \subset R\) such that \(R/J\) has a finite
  \(\C\)-vector space basis (consisting of images of those monomials
  not in \(J\)). By abuse of notation, such monomial ideals are called
  \emph{artinian}, and the \(\C\)-vector space dimension of \(R/J\) is
  called the \emph{colength} of \(J\).

  We get in this way a bijection between
  \begin{enumerate}
  \item integer partitions of \(n\),
  \item order ideals in \((\Nat^2,\le)\) of cardinality \(n\),
  \item monoid ideals in \(\Nat^2\) whose complement has cardinality
    \(n\), and
  \item monomial ideals in \(R\) of colength \(n\).
 \end{enumerate}


 Recall that if \(\mathfrak{a}\) is an ideal in the
 commutative unitary ring \(S\), then the \emph{integral closure}
 \(\bar{\mathfrak{a}}\) consists of all \(u \in S\) that fulfill some
 equation of the form
 \begin{equation}
   \label{eq:iceq}
   s^n + b_1s^{n-1} + \dots + b_n=0, \qquad b_i \in \mathfrak{a}^i
 \end{equation}
 Then \(\mathfrak{a}\) is always contained in its integral closure,
 which is an ideal. The ideal \(\mathfrak{a}\) is said to be
 \emph{integrally closed} if it coincides with its integral closure.

Note that this notion is \textbf{different} from the integral closure 
   of \(\mathfrak{a}\) as a \textbf{subring} of \(S\). On the other
   hand, one can show\footnote{See for instance \cite[chapter 6.6]{Vas:Com}} that 
   the integral closure of the Rees algebra \(S[\mathfrak{a}t]
   \subseteq S[t]\) is equal to the graded subring
   \begin{displaymath}
     S + \bar{\mathfrak{a}}t + \cdots +      \overline{\mathfrak{a}^n}t^n + \cdots
   \end{displaymath}


 For the special case \(S=R\), we have that the integral closure of a
 monomial ideal is again a monomial ideal, and that the latter monomial
 ideal corresponds to the integral closure of the monoid ideal
 corresponding to the former monomial ideal 
\cite{CR:IM, V:MonAlg}. Hence, we have a  bijection between

 \begin{enumerate}
 \item \concave integer partitions of \(n\),
 \item integrally closed monoid ideals in \(\Nat^2\) whose complements
   have cardinality
   \(n\), and
 \item integrally closed monomial ideals in \(R\) of colength \(n\).
 \end{enumerate}

 Fr{\"o}berg and Barucci \cite{Froeberg:Finco}
 studied the growth of the number of ideals of colength \(n\)
 in certain rings, among them local noetherian rings of dimension 1.
 Studying the growth of the number of monomial
 ideals of colength \(n\) in \(R\) is, by the above, the same as
 studying the partition function \(p(n)\). In this article, we will
 instead study the growth of the number of integrally closed monomial
 ideals in \(R\), that is, the number of \concave partitions of \(n\).

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\end{picture}
\caption{\(\mu\) and \(\bar{\mu}\)} \label{fig:exejkonk}
\end{figure} 

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%\caption{\(\mu\) and \(\bar{\mu}\)} \label{fig:exejkonk}
%\end{figure}

\end{section}




\begin{section}{Inequalities defining \concave partitions}
  It is in general a hard problem to compute the integral closure of
  an ideal in a commutative ring. However, for monomial ideals in a
  polynomial ring, the following theorem, which can be found in
  \cite[Exercise 4.23]{Ebud:View} or in \cite{CR:IM, V:MonAlg}, makes
  the problem feasible. 
  \begin{theorem}\label{thm:convex}
    Let \(I \subset \Nat^2\) be a monoid ideal, and regard \(\Nat^2\)
    as a subset of \(\Q^2\) in the natural way. Let
    \(\mathrm{conv}_\Q(I)\) denote the convex hull of \(I\) inside
    \(\Q^2\).
    Then the integral closure of \(I\) is given by
    \begin{equation}
      \label{eq:latticep}
      \mathrm{conv}_\Q(I) \cap \Nat^2
    \end{equation}
  \end{theorem}

\begin{example}
The partition \(\mu=(4,4,2,2)\) corresponds to the monoid ideal
\[((0,4),(2,2),(4,0)),\]
 which has  integral closure
\[((0,4),(1,3),(2,2),(3,1),(4,0)).\]
It follows that \(\overline{\mu}=(4,3,2,1)\).
In figure~\ref{fig:exejkonk} we have drawn the lattice points belonging
to \(F(\mu)\) as dots, and the lattice points belonging to
\(I(\lambda)\) as crosses.
\end{example}


The above theorem gives the following characterization of
\concave partitions:
\begin{lemma}\label{lemma:concaveChar}
  Let \(\lambda=(\lambda_1,\lambda_2,\lambda_3,\dots)\) be a
  partition. Then \(\lambda\) is \concave iff
  for all positive integers \(i &lt; j &lt; k\),
  \begin{equation}
    \label{eq:konkav}
    \lambda_j &lt;  1+ \lambda_i \frac{k-j}{k-i} + \lambda_k
    \frac{j-i}{k-i}
  \end{equation}
  or, equivalently, if
  \begin{equation}
    \label{eq:konkavhom}
    \lambda_i (j-k)  + \lambda_j (k-i) + \lambda_k (i-j) &lt; k-i
  \end{equation}
\end{lemma}

Since all quantities involved are integers, \eqref{eq:konkavhom} is
equivalent to 
\begin{equation}
  \label{eq:konkavhom2}
  \lambda_i (k-j)  + \lambda_j (i-k) + \lambda_k (j-i) \ge i-k+1
\end{equation}

\end{section}



\begin{section}{Generating functions for \pseudoconcave partitions}
We will enumerate \concave partitions by considering another class of
partitions which is more amenable to enumeration, yet is close to
that of \concave partitions.
\begin{definition}\label{def:almostconcave}
  Let \(\lambda=(\lambda_1,\lambda_2,\lambda_3,\dots)\) be a
  partition. Then \(\lambda\) is \emph{\pseudoconcave} iff
  for all positive integers \(i &lt; j &lt; k\),
  \begin{equation}
    \label{eq:pkonkav}
    \lambda_i (k-j)  + \lambda_j (i-k) + \lambda_k (j-i) \ge 0
  \end{equation}
\end{definition}

The reader should note that it is actually a
\emph{stronger} property to be \pseudoconcave than to be \concave. 
Unlike the latter property, it is not
necessarily preserved by conjugation: the partition \((2)\) is
\pseudoconcave, hence \concave, but its conjugate \((1,1)\) is \concave
but not \pseudoconcave.


\begin{theorem}\label{thm:pseudoconcave}
  Let \(\lambda=(\lambda_1,\lambda_2,\lambda_3,\dots)\) be a
  partition, and let \(\mu =(\mu_1,\mu_2,\mu_3,\dots)\) be its
  conjugate, so that \(\card{\setsuchas{j}{\mu_j=i}} = \lambda_i -
  \lambda_{i+1}\) for all \(i\).
  Then the following are equivalent:
  \begin{enumerate}[(i)]
  \item \label{it:ps}
    \(\lambda\) is \pseudoconcave,
  \item \label{it:sv}
    for all positive \(\ell\),
    \begin{equation}
      \label{eq:asc}
            -\lambda_\ell +2\lambda_{\ell+1} - \lambda_{\ell+2} \le 0
    \end{equation}
  \item \label{it:asc}
    for all positive \(\ell\),
    \begin{equation}
      \label{eq:asc2}
            \lambda_{\ell+1} -\lambda_\ell \ge \lambda_{\ell+2} -
            \lambda_{\ell+1}
    \end{equation}
  \item  \label{it:sfp}
    \(\card{\setsuchas{k}{\mu_{k}=i}} \geq
      \card{\setsuchas{k}{\mu_{k}=j}}\) whenever \(i \leq j\).
  \end{enumerate}
\end{theorem}
\begin{proof} \eqref{it:ps} \(\iff\) \eqref{it:sv}:
  Let \(\vek{e}_i\) be the vector with 1 in the \(i\)'th coordinate
  and zeros elsewhere, let \(\vek{f}_j=-\vek{e}_j + 2\vek{e}_{j+1}
  -\vek{e}_{j+2}\), and let \(\vek{t}_{i,j,k} =
    (j-k) \vek{e}_i  +  (k-i)\vek{e}_j  + (j-i)\vek{e}_k\).
  Clearly, \eqref{eq:pkonkav} is equivalent with
  \(\vek{t}_{i,j,k} \cdot \lambda \le 0\), and
  \eqref{eq:asc} is equivalent with
  \(\vek{f}_{j} \cdot \lambda \le 0\).
  We have that \(\vek{f}_{\ell} = \vek{t}_{\ell,\ell+1,\ell+2}\).
  Conversely, we claim that \(\vek{t}_{i,j,k}\)
   is a positive linear combination of different \(\vek{f}_{\ell}\).
  From this claim, it follows
  that if \(\lambda\) fulfills \eqref{eq:asc} for all \(\ell\)
  then \(\lambda\) is \pseudoconcave.

  We can without loss of generality assume that \(i=1\).
  Then it is easy to verify that
  \begin{equation}
    \label{eq:poslincomb}
    \vek{t}_{1,j,k} = \sum_{\ell=1}^{j-2} \ell (k-j)\vek{f}_{\ell}
    + \sum_{\ell=j-1}^{k-2} \ell (j-1)(k-\ell - 1) \vek{f}_{\ell}
  \end{equation}

  \eqref{it:sv} \(\iff\) \eqref{it:asc}    \(\iff\)  \eqref{it:sfp} :
  This is obvious.
\end{proof}


The \emph{difference operator} \(\Delta\) is defined on partitions by
\begin{equation}
  \label{eq:delta}
  \Delta(\lambda_1,\lambda_2,\lambda_3,\dots) = (\lambda_1 -
  \lambda_2, \lambda_2 - \lambda_3, \lambda_3-\lambda_4,\dots)
\end{equation}
We get that the \emph{second order difference operator} \(\Delta^2\)
is given by
\begin{multline}
  \label{eq:delta2}
    \Delta^2(\lambda_1,\lambda_2,\lambda_3,\dots) =   
    \Delta(\Delta(\lambda_1,\lambda_2,\lambda_3,\dots)) = \\
    = (\lambda_1 - 2\lambda_2 + \lambda_3, \lambda_2 - 2\lambda_3 +
    \lambda_4, \lambda_3 - 2\lambda_4 + \lambda_5, \dots)
\end{multline}
\begin{corollary}
  The \pseudoconcave partitions are precisely those with non-negative
  second differences.
\end{corollary}


\begin{definition}
    Let \(p_{sc}(n)\) denote the number of \pseudoconcave partitions of
  \(n\), and \(p_{sc}(n,r)\) denote the number of \pseudoconcave
  partitions of
  \(n\) with at most \(r\) parts.
  Let similarly \(p_{c}(n)\) and \(p_{c}(n,r)\)
denote the number of \concave partitions of \(n\),
and the number of \concave partitions of
\(n\) with at most \(r\) parts, respectively.
  For a partition \(\lambda=(\lambda_1,\lambda_2,\dots)\) let
  \(\vek{x}^\lambda= x_1^{\lambda_1}  x_2^{\lambda_2} \cdots\),
  and define
  \begin{equation}
    \label{eq:PS}
    \begin{split}
    PS(\vek{x}) &amp;= \sum_{\lambda \text{ \pseudoconcave}} \vek{x}^\lambda\\
    PS_r(x_1,\dots,x_r) &amp;= PS(x_1,x_2,\dots,x_r,0,0,0,\dots)
    = \sum_{\substack{\lambda \text{ \pseudoconcave}\\
        \lambda_{r+1}=0}} \vek{x}^\lambda \\
    PC(\vek{x}) &amp;= \sum_{\lambda \text{ \concave}} \vek{x}^\lambda\\
    PC_r(x_1,\dots,x_r) &amp;= PC(x_1,x_2,\dots,x_r,0,0,0,\dots)
    = \sum_{\substack{\lambda \text{ \concave}\\
        \lambda_{r+1}=0}} \vek{x}^\lambda
    \end{split}
  \end{equation}
\end{definition}


Partitions with non-negative second differences have been studied by
Andrews \cite{Andrews:MM2}, who proved that there are as many such
partitions of \(n\) as 
there are partitions of \(n\) into triangular numbers. 


Canfield et al \cite{CCH:RandomPart} have studied partitions with
non-negative \(m\)'th differences. Specializing their results to the
case \(m=2\), we conclude:

\begin{theorem}\label{thm:nonnegsecdiff}
  Let \(n,r\) be  denote positive integers.

  \begin{enumerate}[(i)]
  \item There is a bijection between
    partitions of \(n\) into triangular numbers and \pseudoconcave
    partitions.
  \item     The multi-generating function for \pseudoconcave partitions is
    given by
    \begin{equation}
      \label{eq:psgen}
      \begin{split}
      PS(\vek{x}) &amp;=  \frac{1}{ \prod_{i=1}^\infty \left(1 - \prod_{j=1}^i
        x_j^{1+i-j}\right)}\\
      &amp;=
      1  +
      x_{{1}} +
      {x_{{1}}}^{2} +
      {x_{{1}}}^{3} +
      {x_{{1}}}^{4} +
      {x_{{1}}}^{2}x_{{2}} +
      {x_{{1}}}^{5} +
      {x_{{1}}}^{4}x_{{2}} +
      {x_{{1}}}^{3}x_{{2}}     + \dots
      \end{split}
      \end{equation}
    \item     The multi-generating function for \pseudoconcave partitions with at
    most \(r\) parts is
    given by
    \begin{equation}
      \label{eq:psgenrp}
      PS_r(x_1,x_2,\dots,x_r) =
\frac{1}{ \prod_{i=1}^r \left(1 - \prod_{j=1}^i
        x_j^{1+i-j}\right)}
    \end{equation}
  \item
    The generating function for \pseudoconcave partitions is
    \begin{equation}
      \label{eq:psgenspec}
      PS(t) = \sum_{n=0}^\infty p_{sc}(n)t^n =
      \prod_{i=1}^\infty \frac{1}{ 1 - t^{\frac{i(i+1)}{2}}}
    \end{equation}
    and the one for \pseudoconcave partitions with at most \(r\) parts
    is
    \begin{equation}
      \label{eq:psgenspecrp}
      PS_r(t) = \sum_{n=0}^\infty p_{sc}(n,r)t^n =
      \prod_{i=1}^r \frac{1}{ 1 - t^{\frac{i(i+1)}{2}}}
    \end{equation}

    \item       The proportion of \pseudoconcave partitions with
      at most \(r\) parts among all partitions with at most \(r\) parts
      is
    \begin{equation}
      \label{eq:prop}
      \frac{r!}{\prod_{i=1}^r \frac{i(i+1)}{2}}.
    \end{equation}

  \item As \(n \to \infty\),
      \begin{equation}
        \label{eq:psas}
        \begin{split}
         p_{sc}(n)  \sim c n^{-3/2} \exp(3C n^{1/3})\\
         C = 2^{-1/3} \left[\zeta (3/2)\Gamma(3/2)\right]^{2/3}, \quad
         c = \frac{\sqrt{3}}{12}\,\left ({\frac {C}{\pi }}\right )^{3/2}
        \end{split}
      \end{equation}
  \end{enumerate}
\end{theorem}
  The sequence \(\bigl(p_{sc}(n)\bigr)_{n=0}^\infty\) is identical to
  sequence 
\htmladdnormallink{A007294}{http://www.research.att.com/projects/OEIS?Anum=007294}
in OEIS \cite{Sloane}.
    We have submitted the sequences
    \(\bigl(p_{sc}(n,r)\bigr)_{n=0}^\infty\), for \(r=3,4\), in
    OEIS \cite{Sloane}, as 
\htmladdnormallink{A086159}{http://www.research.att.com/projects/OEIS?Anum=086159}
and 
\htmladdnormallink{A086160}{http://www.research.att.com/projects/OEIS?Anum=086160}.
The sequence for \(r=2\) was already in the database, as 
\htmladdnormallink{A008620}{http://www.research.att.com/projects/OEIS?Anum=008620}. 


\begin{subsection}{Other appearances of \pseudoconcave partitions in
  the literature} 
  The bijection between partitions into triangular numbers and
  partitions with non-negative second difference  is mentioned in
  \htmladdnormallink{A007294}{http://www.research.att.com/projects/OEIS?Anum=007294}  
in OEIS \cite{Sloane}, together with a reference to Andrews
\cite{Andrews:MM2}.  That sequence has been contributed by  Mira
  Bernstein and Roland Bacher; 
  we thank Philippe Flajolet for drawing our attention to it.




Gert Almkvist \cite{Almkvist:partitions} gives an asymptotic
analysis of \(p_{sc}(n)\) which is finer than \eqref{eq:psas}.

Another derivation of the generating functions above can found in a
forthcoming paper ``Partition Bijections, a Survey'' \cite{Pak:Survey}
by Igor Pak. He observes that the set of 
\pseudoconcave  partitions with at most \(r\) parts consists of the
lattice points of the unimodular cone spanned by the vectors
\(v_0=(1,\dots,1)\) and \(v_i =(i-1,i-2,\dots,1,0,0,\dots)\) for \(1
\le i \le r\). 

Corteel and Savage \cite{PCInEq} calculate rational generating functions for
classes of partitions defined by linear homogeneous inequalities. This
applies to \pseudoconcave partitions, but not directly to concave partitions,
since the inequalities \eqref{eq:konkavhom} defining them are inhomogeneous.
\end{subsection}
\end{section}








    \begin{section}{Generating functions for \concave partitions}

      Recall that a \concave partition
      \(\vek{\lambda}=(\lambda_1,\lambda_2,\dots)\) fulfills
      \eqref{eq:konkavhom2}, and that conversely, every sequence of
      non-negative integers which is eventually zero and fulfills
      \eqref{eq:konkavhom2} gives a \concave partition. If we fix a
      positive integer \(r\), then we need only finitely many of the
      inequalities in \eqref{eq:konkavhom2}: we can take those indexed
      by \(i &lt; j &lt; k &lt; r+2\), together with the non-negativity
      conditions \(\lambda_i \ge 0\).
      Hence, there is a matrix
      \(A\) with \(r\) columns, and whose rows are indexed by tuples
      \((i,j,k)\) with \(k \le r+1\), so that
      a \concave partition with at most \(r\) parts
      corresponds to a solution to
      \begin{equation}
        \label{eq:b}
        A\vek{\lambda} \ge \vek{b}, \qquad \vek{\lambda} \in \Nat^r,
      \end{equation}
      whereas a \pseudoconcave partition with at most \(r\) parts
      corresponds to a solution to
      \begin{equation}
        \label{eq:A}
        A\vek{\lambda} \ge \vek{0}, \qquad \vek{\lambda} \in \Nat^r.
      \end{equation}


      We let \(\mathcal{K} = \setsuchas{\alpha \in \R^r}{A\alpha \ge \vek{b},
        \vek{\alpha} \ge \vek{0}}\), 
      \(\mathcal{P} = \setsuchas{\alpha \in \R^r}{A\alpha \ge \vek{0},
        \vek{\alpha} \ge \vek{0}}\). Then \(\mathcal{P}\) is a rational
      polyhedron in the positive orthant. Since the RHS vector
      \(\vek{b}\) is non-positive, \(\mathcal{P}\) contains its recession
      cone \(\mathcal{K}\). The solutions to \eqref{eq:A} and
      \eqref{eq:b} are precisely \(\mathcal{KI}=\mathcal{K} \cap \Nat^r\) and
      \(\mathcal{PI}=\mathcal{P} \cap \Nat^r\), and the generating functions of
      these two sets of lattice points are precisely \(PS_r\) and \(PC_r\).

      \begin{example}
      If \(r=3\) and if we order the 3-subsets of
      \(\{1,2,3,4\}\) as \(123\), \(124\), \(134\), \(234\) then
      \begin{displaymath}
        A =
        \begin{pmatrix}
          1 &amp; -2 &amp; 1  \\
          2 &amp; -3 &amp; 0  \\
          1 &amp; 0 &amp; -3 \\
          0 &amp; 1 &amp; -2 
        \end{pmatrix}, 
        \qquad 
        \vek{b} = (-1,-2,-2,-1)^t.
      \end{displaymath}
      \(\mathcal{K}\) is the cone generated by the rays \((1,0,0)\),
      \((2,1,0)\), and \((3,2,1)\), whereas \(\mathcal{P}\)
      is the Minkowski sum of \(\mathcal{K}\) and the
      polytope which is the convex hull of 
      \((0,0,0)\),
      \(( 0,   0, 1/2\),
      \(( 0, 1/3, 2/3)\),
      \(( 0, 1/2,   0)\),
      \(( 0, 2/3, 1/3)\),
      \(( 0, 2/3, 2/3)\). So \(\mathcal{P}\) is a rational polyhedron
      but not a lattice polyhedron.
      \end{example}



      \begin{lemma}\label{lemma:sameden}
        The generating function \(PC_r(x_1,\dots,x_r)\) is a rational
        function with the same denominator as \(PS_r(x_1,\dots,x_r)\),
        and with a numerator which evaluates to 1 at
        \((1,\dots,1)\). In other words,
        \begin{equation}\label{eq:PCr}
          PC_r(x_1,\dots,x_r) =
          \frac{Q_r(x_1,\dots,x_r)}
          {\prod_{i=1}^r \left( 1 - \prod_{j=1}^i
              x_j^{1+i-j}\right)}, \quad  Q_r(1,\dots,1)=1.
        \end{equation}
      \end{lemma}
      \begin{proof}
        This can be obtained from the corresponding result for linear
        diophantine \emph{equalities}\footnote{See \cite[Corollary
        3.8]{Stanley:CombCom} and the paragraph immediately following
        it} by adding slack-variables and then specializing the
        corresponding formal variables to 1. We give the outline of a
        self-contained proof.
        
      By Gordan's lemma\footnote{Gordan's lemma says the lattice
        points in a finitely generated rational cone in the positive
        orthant constitute a normal affine semigroup, see
        \cite[Proposition 6.1.2]{BrunsHerzog:CM}}, \(\mathcal{KI}\) is a finitely
      generated affine semigroup. In fact, it has a unique finite minimal
      generating set, called its \emph{Hilbert basis}. 
      Furthermore, \(\mathcal{PI}\)  is a
      module over \(\mathcal{KI}\), by which we mean that
      \(\mathcal{KI} + \mathcal{PI} \subseteq \mathcal{PI}\). 
      Now let \(R=\C[\mathcal{KI}]\) be the semigroup ring on
      \(\mathcal{KI}\), i.e. the \(\C\)-vector space spanned by all
      monomials \(\setsuchas{\vek{x}^\vek{\alpha}}{\vek{\alpha} \in
        \mathcal{KI}}\).
      We define 
      \begin{equation}
        \label{eq:monmult}
\vek{x}^\vek{\alpha}
      \vek{x}^\vek{\beta} = \vek{x}^{\vek{\alpha} + \vek{\beta}},        
      \end{equation}
      and
      extend this multiplication by linearity to all of \(R\), turning
      it into a \(r\)-multigraded, noetherian  \(\C\)-algebra. 
      Similarly, we define \(M\) to be the 
      \(\C\)-linear span of monomials corresponding to points in \(\mathcal{PI}\).
      \eqref{eq:b}. The multiplication \eqref{eq:monmult} gives \(M\)
      the structure of \(r\)-multigraded \(R\)-module. 
      
      Since \(\mathcal{KI}\) is a finitely    generated affine
      semigroup,  \(R\) is a  finitely generated \(\C\)-algebra. Since
      it is a subring of \(\C[x_1,\dots,x_r]\), it is  an integral domain.
      The Hilbert series of \(R\) is \(PS_r\), and the Hilbert series
      of \(M\) is \(PC_r\).   

      Now note that since 
      there is some \(\gamma \in \Nat^r\) such that \(\mathcal{PI} +
      \gamma \subseteq \mathcal{KI}\), it follows that \(M\) is isomorphic as an
      \(R\)-module to the ideal \(\vek{x}^\gamma M \subseteq
      R\). Consequently, \(M\) is
      a finitely generated torsion-free module over \(R\), of rank
      1. Its annihilator is 
      zero, so  \(M\) has the same dimension as \(R\). 
      

      It follows from standard commutative
      algebra\footnote{See  \cite[Theorem 2.3]{Stanley:CombCom},
      and note that \(M\) is \(\Nat^r\)-graded rather than
      \(\Z^r\)-graded} that the Hilbert series of \(R\) and \(M\)
      are rational, of the form 
      \begin{displaymath}
        \frac{N_R(x_1,\dots,x_r)}
        {\prod_{i=1}^s     (1-\vek{x}^{\vek{\alpha}_i})} 
      \qquad \text{ and } \qquad
      \frac{N_M(x_1,\dots,x_r)}
      {\prod_{i=1}^s       (1-\vek{x}^{\vek{\alpha}_i})},
      \end{displaymath}
      where the \(\vek{\alpha}_i\)'s are the elements of a
      basis of \(\mathcal{KI}\), and the polynomials \(N_R\) and
      \(N_M\) have rational coefficients. Since we know the Hilbert
      series of \(R\), we conclude that the vectors
      \((i,i-1,\dots,0,\dots,0)\), for \(1 \le i \le r\), form a 
      basis for \(\mathcal{KI}\).

      Furthermore\footnote{See
      \cite{BrunsHerzog:CM} , exercise 4.4.12}, 
      \(N_R(1,\dots,1)=1\) , and \(N_M(1,\dots,1) = \textrm{rank}(M)=1\).
      The ring \(R\) is Cohen-Macaulay, hence\footnote{See  again
      \cite{BrunsHerzog:CM} , exercise 4.4.12}  all coefficients 
      of \(N_R(t,\dots,t)\) are non-negative. As calculated in
      \eqref{eq:Qrt}, the polynomials \(N_M(t,\dots,t)\) have some negative
      coefficients for \(r=2,3,4\), so \(M\) is not Cohen-Macaulay in
      general.

    \end{proof}







    We can say something more about the numerators:
    \begin{theorem}\label{thm:pcoco}
      Let \(r\) be a fixed positive integer. Then
      \begin{enumerate}[(A)]
      \item The multigenerating function of \concave partitions with at
        most \(r\) parts is given by \eqref{eq:PCr}, 
      where \(Q_r(x_1,\dots,x_r)\) is a polynomial with
      integer coefficients such that
      all exponent vectors of the monomials that occur
      in \(Q_r\) are weakly decreasing.
      



\item
 The generating function for \concave partitions with at most
      \(r\) parts is given by
      \begin{equation}
        \label{eq:concgenf}
        PC_r(t) =  \sum_{n=0}^\infty p_{c}(n,r)t^n =
       \frac{Q_r(t)}{\prod_{i=1}^r \left( 1 -
           t^{\frac{i(i+1)}{2}}\right)}
      \end{equation}
      where \(Q_r(1)=1\), and the numerator has degree strictly smaller  than
      \(r^3/6 + r^2/2 + r/3\).


\item \(p_{c}(n,r) \sim p_{sc}(n,r)\) as \(n \to \infty\).

    \item       The proportion of concave partitions with
      at most \(r\) parts among all partitions with at most \(r\) parts
      is the same as the proportion of  \pseudoconcave partitions with
      at most \(r\) parts among all partitions with at most \(r\)
      parts, namely
    \begin{equation}
      \label{eq:prop2}
      \frac{r!}{\prod_{i=1}^r \frac{i(i+1)}{2}}.
    \end{equation}


  \item \(Q_r(x_1,\dots,x_r)=Q_{r+1}(x_1,\dots,x_r,0)\).

  \item
      \begin{equation}
        \label{eq:PC}
        PC(\vek{x}) =
        \frac{Q(\vek{x})}
        {\prod_{i=1}^\infty \left( 1 - \prod_{j=1}^i
            x_j^{1+i-j}\right)}
      \end{equation}
      where \(Q(\vek{x})\) is a formal power series with the property
      that for each \(\ell\),
      \(Q(x_1,\dots,x_\ell,0,0,\dots)=Q_\ell(x_1,\dots,x_\ell)\); in
      other words,
      \begin{displaymath}
        Q = 1 + \sum_{i=1}^\infty \left(Q_i - Q_{i-1}\right)
      \end{displaymath}

\end{enumerate}
        \end{theorem}

    \begin{proof}

    All monomials in
      \begin{displaymath}
        \prod_{i=1}^r \left( 1 - \prod_{j=1}^i
        x_j^{1+i-j}\right)
      \end{displaymath}
      have weakly decreasing exponent vectors, as have all monomials in
      the power series \(PC_r(x_1,\dots,x_r)\). Summing  weakly
      decreasing exponent vectors gives  weakly decreasing exponent
      vectors, so all exponent vectors in \(Q_r(x_1,\dots,x_r)\) are
      weakly decreasing.

      

      



      If we specialize \(x_1=x_2=\dots = x_r = t\) we get 
      \begin{displaymath}
        PC_r(t) =  
       \frac{Q_r(t)}{\prod_{i=1}^r \left( 1 -
           t^{\frac{i(i+1)}{2}}\right)}, \qquad
       PS_r(t) = \frac{1}{\prod_{i=1}^r \left( 1 -
           t^{\frac{i(i+1)}{2}}\right)}.
      \end{displaymath}


      Thus \(Q_r(1) =1\),
      and we  conclude that
      \(p_{c}(n,r) \sim p_{sc}(n,r)\) as \(n \to
      \infty\).

      Furthermore, from Stanley's ``grey book''
      \cite[Theorem 4.6.25]{Stanley:En1} we have that the rational
      function \(PC_r(t,\dots,t)\) is of degree \(&lt;0\). The degree of
      the denominator is 
      \begin{displaymath}
        \sum_{i=1}^r \frac{i(i+1)}{2} = 
        \frac{r^3}{6} + \frac{r^2}{2} + \frac{r}{3}
      \end{displaymath}
      so \(Q_r(t)\) have smaller degree than that.


      If \((\lambda_1,\dots,\lambda_r,\lambda_{r+1})\) is a 
      \concave  partition, then so is
      \((\lambda_1,\dots,\lambda_r,0)\);
      it follows that \(Q_{r+1}(x_1,\dots,x_r,0) =Q_r(x_1,\dots,x_r)\).
      The assertion about \(PC(\vek{x})\) follows by passing to the
      limit.
    \end{proof}



By generating all concave partitions of \(n\) with at most \(r\)
parts, up to a large \(n\), we have calculated that
    \begin{equation}
      \label{eq:Qr}
      \begin{split}
      Q_1(\vek{x}) &amp;= 1 \\
      Q_2(\vek{x}) &amp; =1 + x_1x_2 - x_1^2x_2\\
      Q_3(\vek{x}) &amp; =Q_2(\vek{x})  +  x_3\left({x_{{1}}}^{5}{x_{{2}}}^{3} -
        {x_{{1}}}^{4}{x_{{2}}}^{3} - 2\,{x_{{1}}}^{3}
{x_{{2}}}^{2} + {x_{{1}}}^{2}{x_{{2}}}^{2} + x_{{1}}x_{{2}}
 \right)
      \end{split}
    \end{equation}
and that
    \begin{equation}
      \label{eq:Qrt}
      \begin{split}
        Q_1(t)  &amp;= 1\\
        Q_2(t)  &amp; = 1+t^2-t^3\\
        Q_3(t) &amp;= 1+t^2+t^5-2t^6-t^8+t^9\\
        Q_4(t) &amp;=
        1 + t^2 + t^4 + t^5-t^6-t^7 + 2t^9-2t^{10}-t^{11}-2t^{12} +
        \\ &amp;+
        2t^{13}-t^{14}-t^{15} + t^{16} + t^{17} + t^{18}-t^{19}
      \end{split}
    \end{equation}
    We have also used the package LinDiophanthus \cite{LinDiophantus} by Doron
    Zeilberger to verify our results. 


By generating all concave partitions of \(n\) for \(n \le 20\)
we have calculated that
    \begin{multline}
      \label{eq:PC2}
      PC(t)=\sum_{n=0}^\infty p_c(n)t^n =
      1+ t + 2 t^2 + 3 t^3 + 4 t^4 + 7 t^5 + 9 t^6 + 11 t^7 +
      \\ +17t^8  +23 t^9 + 28 t^{10} + 39 t^{11} + 48 t^{12}
      + 59 t^{13} + 79 t^{14} +
      \\ +      100 t^{15} + 121 t^{16}
      +152 t^{17} + 185 t^{18} + 225 t^{19} + 280 t^{20} + O(t^{21})
    \end{multline}
    Based on \eqref{eq:Qrt}, we conjecture that
    \begin{equation}
      \label{eq:PC3}
      PC(t) = \frac{1 + t^2 + O(t^3)}{\prod_{i=1}^\infty \left( 1 -
           t^{\frac{i(i+1)}{2}}\right)}
    \end{equation}
    We also conjecture that \(\log p_c(n)\) grows as
    \(n^{1/3}\), i.e. approximately as fast as \pseudoconcave
    partitions.

    The sequences \(\bigl(p_c(n)\bigr)_{n=0}^\infty\) are in the OEIS
    \cite{Sloane} as 
    \htmladdnormallink{A084913}{http://www.research.att.com/projects/OEIS?Anum=084913}.
    The sequences \(\bigl(p_c(n,r)\bigr)_{n=0}^\infty\)
    are 
    \htmladdnormallink{A086161}{http://www.research.att.com/projects/OEIS?Anum=086161},   
    \htmladdnormallink{A086162}{http://www.research.att.com/projects/OEIS?Anum=086162},   
    and 
    \htmladdnormallink{A086163}{http://www.research.att.com/projects/OEIS?Anum=086163}   
    for \(r=2,3,4\).



\end{section}


\begin{thebibliography}{10}

\bibitem{Almkvist:partitions}
Gert Almkvist.
\newblock Asymptotics of various partitions.
\newblock manuscript.

\bibitem{Andrews:MM2}
George~E. Andrews.
\newblock Mac{M}ahon's partition analysis. {II}. {F}undamental theorems.
\newblock {\em Ann. Comb.}, 4(3-4):327--338, 2000.
\newblock Conference on Combinatorics and Physics (Los Alamos, NM, 1998).

\bibitem{Froeberg:Finco}
Valentina Barucci and Ralf Fr{\"o}berg.
\newblock On the number of ideals of finite colength.
\newblock In {\em Geometric and combinatorial aspects of commutative algebra
  (Messina, 1999)}, volume 217 of {\em Lecture Notes in Pure and Appl. Math.},
  pages 11--19. Dekker, New York, 2001.

\bibitem{BrunsHerzog:CM}
Winfried Bruns and J{\"u}rgen Herzog.
\newblock {\em Cohen-{M}acaulay rings}.
\newblock Cambridge University Press, Cambridge, 1993.

\bibitem{CCH:RandomPart}
Rod Canfield, Sylvie Corteel, and Pawel Hitczenko.
\newblock Random partitions with non-negative {$r$}-th differences.
\newblock {\em Adv. in Appl. Math.}, 27(2-3):298--317, 2001.
\newblock Special issue in honor of Dominique Foata's 65th birthday
  (Philadelphia, PA, 2000).

\bibitem{PCInEq}
S.~Corteel and C.~D. Savage.
\newblock Partitions and compositions defined by inequalities.
\newblock math.CO/0309110.

\bibitem{Ebud:View}
David Eisenbud.
\newblock {\em Commutative {A}lgebra with a {V}iew {T}oward {A}lgebraic
  {G}eometry}, volume 150 of {\em Graduate {T}exts in {M}athematics}.
\newblock Springer {V}erlag, 1995.

\bibitem{Pak:Survey}
Igor Pak.
\newblock Partition bijections, a survey.
\newblock {\em Ramanujan Journal}, to appear.

\bibitem{CR:IM}
Veronica~Crispin Quinonez.
\newblock Integrally closed monomial ideals and powers of ideals.
\newblock Technical report, Stockholm University, 2002.

\bibitem{Sloane}
Neil J.~A. Sloane.
\newblock The on-line encyclopedia of integer sequences.
\newblock {\url{http://www.research.att.com/~njas/sequences/index.html}}.

\bibitem{Stanley:CombCom}
Richard~P. Stanley.
\newblock {\em Combinatorics and {C}ommutative {A}lgebra}, volume~41 of {\em
  Progress in {M}athematics}.
\newblock Birkh{\"a}user, 2 edition, 1996.

\bibitem{Stanley:En1}
Richard~P. Stanley.
\newblock {\em Enumerative combinatorics. {V}ol. 1}.
\newblock Cambridge University Press, Cambridge, 1997.
\newblock With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986
  original.

\bibitem{Vas:Com}
Wolmer~V. Vasconcelos.
\newblock {\em Computational {M}ethods in {C}ommutative {A}lgebra and
  {A}lgebraic {G}eometry}.
\newblock Algorithms and {C}omputations in {M}athematics. Springer Verlag,
  1998.

\bibitem{V:MonAlg}
R.~Villarreal.
\newblock {\em Monomial {A}lgebras}.
\newblock Marcel {D}ekker, 2001.

\bibitem{LinDiophantus}
Doron Zeilberger.
\newblock Lindiophantus.
\newblock \url{http://www.math.temple.edu/~zeilberg/}, 2001.
\newblock Program for calculating multi-generating functions for solution sets
  of systems of diophantine equations.

\end{thebibliography}


\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}:
Primary 05A17; Secondary 13B22.

\noindent \emph{Keywords:}
integer partitions, monomial ideals, integral closure.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A007294},
\seqnum{A086159},
\seqnum{A086160},
\seqnum{A008620},
\seqnum{A084913},
\seqnum{A086161},
\seqnum{A086162},
\seqnum{A086163}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received October 6 2003;
revised version received February 11 2004.  
Published in {\it Journal of Integer Sequences}, February 11 2004.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}.
\vskip .1in


\end{document}

                                                                                

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