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 Pt(q) := ∑ n≥1 p(n, t) q . Somewhat surprisingly, Pt(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. 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; see, e.g., [2]. A partition of a positive integer n is, as usual, an integer k-tuple λ1 ≥ λ2 ≥ · · · ≥ λk > 0, for some k, such that n = λ1 + λ2 + · · ·+ λk . The integers λ1, λ2, . . . , λk are the parts of the partition. We are interested in the counting function p(n, t) := #partitions of n with difference t between largest and smallest parts. It is immediate that p(n, 0) = d(n) where d(n) denotes the number of divisors of n. Charmingly, p(n, 1) equals the number of nondivisors of n: p(n, 1) = n− d(n) , which can be explained bijectively by the fact that the partitions counted by p(n, 0)+p(n, 1) contain exactly one sample with k parts, for each k = 1, 2, . . . , n [1, Sequence A049820], or by the generating function identity∑ n≥1 p(n, 1) q = ∑ m≥1 qm 1− qm qm+1 1− qm+1 = q (1− q)2 − ∑ m≥1 qm 1− qm . (The last equation follows from a few elementary operations on rational functions). An even less obvious instance of our partition counting function is (1) p(n, 2) = (⌊n 2 ⌋ 2 ) , as observed by Reinhard Zumkeller in 2004 [1, Sequence A008805]. (It is not clear to us where in the literature this formula first appeared, though specific values of p(n, k) are well represented in Date: 14 July 2014. 2010 Mathematics Subject Classification. Primary 11P84; Secondary 05A17.