Partitions with fixed differences between largest and smallest parts

Abstract: Enumeration results on integer partitions form a classic body of mathematics going back to at least Euler, including numerous applications throughout mathematics and some areas of physics. We study the number $p(n,t)$ of partitions of $n$ with difference $t$ between largest and smallest parts. For example, $p(n,0)$ equals the number of divisors of $n$, the function $p(n,1)$ counts the nondivisors of $n$, and $p(n,2) = \binom{ \left\lfloor \frac n 2 \right\rfloor }{ 2 }$. Beyond these three cases, the existing literature contains few results about $p(n,t)$, even though concrete evaluations of this partition function are featured in several entries of Sloane's Online Encyclopedia of Integer Sequences.

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 $t>1$ (e.g., the above formula for $p(n,2)$ is an example of a quasipolynomial with period 2). Our result generalizes to partitions with an arbitrary number of specified distances.

This is joint work with George Andrews (Penn State) and Neville Robbins (SF State).

number theory

Audience: researchers in the topic

EIMI Number Theory Seminar

Series comments: Password: the number of quadratic nonresidues modulo 23