Enumeration of odd-dimensional partitions modulo 4

Abstract

The number of standard Young tableaux on a partition shape \(\lambda\) is called the dimension of the partition. Partitions with odd dimensions were enumerated by McKay and were further characterized by Macdonald in the 1970s. Let \(a_i(n)\) be the number of partitions of \(n\) with dimension congruent to \(i\) modulo \(4\). In this paper, we refine Macdonald's and McKay's results by computing \(a_1(n)\) and \(a_3(n)\) when \(n\) has no consecutive 1s in its binary expansion or when the sum of binary digits of \(n\) is \(2\).