Enumeration of Even Dimensional Partitions modulo 4

TL;DR

The paper addresses the problem of enumerating partitions of $n$ whose SYT dimension $f^{\lambda}$ is congruent to $2$ modulo $4$, extending Macdonald's 2-core tower framework. It develops explicit recurrences for $a_2(n)$ and provides a closed form in the sparse-binary case, together with a general recursive algorithm for all $n$. The approach leverages 2-core towers and a detailed combinatorial analysis of 2-cores and 2-quotients to count partitions by their dimension modulo $4$, building on prior results for odd partitions. The results illuminate the structure of even-dimensional partitions and have implications for the study of spinorial representations of the symmetric group, offering practical tools for computation and potential generalizations to higher powers of two. Overall, the work extends the core-quotient method to a new congruence class and supplies both explicit formulas and recursive strategies for $a_2(n)$.

Abstract

The number of standard Young tableaux possible of shape corresponding to a partition $λ$ is called the dimension of the partition and is denoted by $f^λ$. 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 $n$.

Theorems & Definitions (20)

• Theorem 1
• Corollary 2
• Definition 3
• Example 4
• Example 5
• Lemma 6
• proof
• Example 7
• Example 8
• Example 10