A Note on the Number of $t$-Core Partitions

Abstract

A partition of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. A Ferrers graph represents a partition in the natural way. Fix a positive integer t. A partition of n is called a t−core partition of n if none of its hook numbers are multiples of t. Let ct(n) denote the number of t−core partitions of n. It has been conjectured that if t ≥ 4, then ct(n) > 0 for all n ≥ 0. In [7], the author proved the conjecture for t ≥ 4 even and for those t divisible by at least one of 5, 7, 9, or 11. Moreover if t ≥ 5 is odd, then it was shown that ct(n) > 0 for n sufficiently large. In this note we show that if k ≥ 2, then c3k(n) > 0 for all n using elementary arguments. A partition of a positive integer n is a nonincreasing sequence of positive integers with sum n. Here we define a special class of partitions. Definition 1. Let t ≥ 1 be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t−core partition of n. The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3,4,5]. If t ≥ 1 and n ≥ 0, then we define ct(n) to be the number of partitions of n that are t−core partitions. The arithmetic of ct(n) is studied in [3,4]. The power series generating function for ct(n) is given by the infinite product: (1) ∞ ∑