Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Unimodality of the Andrews-Garvan-Dyson cranks of partitions   The main objective of this paper is to investigate the distribution of the
Andrews-Garvan-Dyson crank of a partition. Let $M(m,n)$ denote the number of
partitions of $n$ with the Andrews-Garvan-Dyson crank $m$, we show that the
sequence \break $\{M(m,n)\}_{|m|\leq n-1}$ is unimodal for $n\geq 44$. It turns
out that the unimodality of \break $\{M(m,n)\}_{|m|\leq n-1}$ is related to the
monotonicity properties of two partition \break functions $p_k(n)$ and
$pp_k(n)$. Let $p_k(n)$ denote the number of partitions of $n$ with at most $k$
parts such that the largest part appears at least twice and let $pp_k(n)$
denote the number of pairs $(\alpha,\beta)$ of partitions of $n$, where
$\alpha$ is a partition counted by $p_k(i)$ and $\beta$ is a partition counted
by $p_{k+1}(n-i)$ for $0\leq i\leq n$. We show that $p_k(n)\geq p_k(n-1)$ for
$k\geq 5$ and $n\geq 14$ and $pp_k(n)\geq pp_k(n-1)$ for $k\geq 3$ and $n\geq
2$. With the aid of the monotonicity properties on $p_k(n)$ and $pp_k(n)$, we
show that $M(m,n)\geq M(m,n-1)$ for $n\geq 14$ and $ 0\leq m \leq n-2$ and
$M(m-1,n)\geq M(m,n)$ for $n\geq 44$ and $1\leq m\leq n-1$. By means of the
symmetry $M(m,n)=M(-m,n)$, we find that $M(m-1,n)\geq M(m,n)$ for $n\geq 44$
and $1\leq m\leq n-1$ implies that the sequence $\{M(m,n)\}_{|m|\leq n-1}$ is
unimodal for $n\geq 44$. We also give a proof of an upper bound for ospt(n)
conjectured by Chan and Mao in light of the inequality $M(m-1,n)\geq M(m,n)$
for $n\geq 44$ and $0\leq m\leq n-1$.

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The categories of this paper are: math-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: q-bio
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The categories of this paper are: comp-gas
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The categories of this paper are: eess
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
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The categories of this paper are: cond-mat
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg