New congruences for t-core partitions and Andrews' singular overpartitions

Abstract

A partition of n is called a t-core partition of n if none of its hook numbers are multiples of t. Let the number of t-core partitions of n be denoted by at(n). Recently, G. E. Andrews defined combinatorial objects which he called (k,i) singular overpartitions, overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. Let the number of (k,i) singular overpartitions of n be denoted by C‾k,i(n). The object of this paper is to obtain new congruences modulo 2 for a15(n) and a23(n). We also obtain congruences modulo 2 for C‾92,23 and C‾60,15.