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 $n \geq 1$, \[ p_{1,5}(m,5n)\geq p_{2,5}(m,5n), \,\,\,1 \leq m \leq 5n. \]