Partitions with fixed differences between largest and smallest parts

TL;DR

This paper derives an explicit rational generating function for the number of partitions of n with a fixed difference t between largest and smallest parts, revealing that these counts are quasipolynomials in n for t>1.

Contribution

It provides a new explicit formula for the generating function of partitions with fixed part differences, generalizing previous results and showing rationality for t>1.

Findings

$P_t(q)$ is rational for $t>1$

$p(n,t)$ is a quasipolynomial in $n$ for fixed $t>1$

Generalizes to partitions with multiple specified distances

Abstract

We study the number $p(n,t)$ of partitions of $n$ with difference $t$ between largest and smallest parts. Our main result is an explicit formula for the generating function $P_t(q) := \sum_{n \ge 1} p(n,t) \, q^n$. Somewhat surprisingly, $P_t(q)$ is a rational function for $t>1$; equivalently, $p(n,t)$ is a quasipolynomial in $n$ for fixed $t>1$. Our result generalizes to partitions with an arbitrary number of specified distances.