Enumeration of Partitions modulo 4

Aditya Khanna

Published 2022-07-15Version 1

The number of standard Young tableaux possible of shape corresponding to a partition $\lambda$ is called the dimension of the partition and is denoted by $f^{\lambda}$. Partitions with odd dimensions were enumerated by McKay and were further classified by Macdonald. 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 calculating $a_1(n)$ 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 and providing values for $a_2(n)$ for all $n$. We also present similar results for irreducible representations of alternating groups.