Partitions with fixed differences between largest and smallest parts

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.

5 pages