Refinements of Some Partition Inequalities

TL;DR

This paper investigates specific partition inequalities related to modular conditions, establishing non-negativity results for certain coefficients and providing combinatorial interpretations involving partitions with parts in specific residue classes.

Contribution

It introduces new inequalities for partition functions based on modular restrictions and proves their non-negativity, extending the understanding of partition inequalities.

Findings

Proved non-negativity of coefficients c(m, Mn) for certain partition generating functions.

Established inequalities between partition counts p_{1,5}(m,5n) and p_{2,5}(m,5n).

Provided combinatorial interpretations for the inequalities in terms of partitions with parts modulo 5.

Abstract

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