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 f t ≥ 4, then ct(n) ≥ 0 for all n ≥ 0. In [7], the author proved the conjecture for f ≥ 4 even and for those t divisible by at least one of 5, 7, 9, or 11. Moreover if f ≥ 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. © 1995 Rocky Mountain Mathematics Consortium.