Enumeration of Odd Dimensional Partitions modulo 4

TL;DR

This paper refines previous results on counting partitions with odd dimensions by explicitly computing the number of such partitions modulo 4 under specific binary expansion conditions.

Contribution

It provides explicit formulas for the counts of partitions with dimensions congruent to 1 or 3 modulo 4 for numbers with particular binary properties.

Findings

Computed $a_1(n)$ and $a_3(n)$ for specific binary conditions.

Extended McKay and Macdonald's enumeration results.

Refined understanding of partition dimensions modulo 4.

Abstract

The number of standard Young tableaux of shape 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 characterized 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 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.