Enumeration of Even Dimensional Partitions modulo 4

TL;DR

This paper extends the enumeration of partitions with odd dimensions to those with dimensions congruent to 2 modulo 4, providing explicit formulas and recursive methods for their enumeration.

Contribution

It introduces a new enumeration for partitions with dimensions ≡ 2 mod 4 using 2-core tower theory, extending previous work on odd dimensions.

Findings

Explicit formulas for a_2(n) when n has no consecutive 1s in binary

Recursive formula for computing a_2(n) for all n

Extension of enumeration from odd dimensions to dimensions ≡ 2 mod 4

Abstract

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 characterized by Macdonald using the theory of 2-core towers. We use the same theory to extend the results to partitions of $n$ with dimensions congruent to 2 modulo 4 which are enumerated by $a_2(n)$. We provide explicit results for $a_2(n)$ when $n$ has no consecutive 1s in its binary expansion and give a recursive formula to compute $a_2(n)$ for all $n$.