11-Regular partitions and a Hecke eigenform

Abstract

A partition of a positive integer [Formula: see text] is called [Formula: see text]-regular if none of its parts is divisible by [Formula: see text]. Let [Formula: see text] denote the number of 11-regular partitions of [Formula: see text]. In this paper we give a complete description of the behavior of [Formula: see text] modulo [Formula: see text] when [Formula: see text] in terms of the arithmetic of the ring [Formula: see text]. This description is obtained by relating the generating function for these values of [Formula: see text] to a Hecke eigenform, and as a byproduct we find exact criteria for which of these values are divisible by 5 in terms of the prime factorization of [Formula: see text].