Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Refinements of Some Partition Inequalities   In the present paper we initiate the study of a certain kind of partition
inequality, by showing, for example, that if $M\geq 5$ is an integer and the
integers $a$ and $b$ are relatively prime to $M$ and satisfy $1\leq a<b<M/2$,
and the $c(m,n)$ are defined by \[
\frac{1}{(sq^a,sq^{M-a};q^M)_{\infty}}-\frac{1}{(sq^b,sq^{M-b};q^M)_{\infty}}:=\sum_{m,n\geq
0} c(m,n)s^m q^n, \] then $c(m, Mn)\geq 0$ for all integers $m\geq 0, n\geq 0$.
%If, in addition, $M$ is even, then $c(m, Mn+M/2)\geq 0$ for all integers
$m\geq 0, n\geq 0$. A similar result is proved for the integers $d(m,n)$
defined by \[
(-sq^a,-sq^{M-a};q^M)_{\infty}-(-sq^b,-sq^{M-b};q^M)_{\infty}:=\sum_{m,n\geq 0}
d(m,n)s^m q^n. \] In each case there are obvious interpretations in terms of
integer partitions. For example, if $p_{1,5}(m,n)$ (respectively
$p_{2,5}(m,n)$) denotes the number of partitions of $n$ into exactly $m$ parts
$\equiv \pm 1 (\mod 5)$ (respectively $\equiv \pm 2 (\mod 5)$), then for each
integer $n \geq 1$, \[ p_{1,5}(m,5n)\geq p_{2,5}(m,5n), \,\,\,1 \leq m \leq 5n.
\]

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The categories of this paper are: math-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: solv-int
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The categories of this paper are: hep-th
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: q-alg
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The categories of this paper are: econ
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The categories of this paper are: physics