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Enumeration of Odd Dimensional Partitions modulo 4

Aditya Khanna

TL;DR

This paper advances the enumeration of partitions by the dimension modulo 4, refining earlier results of McKay and Macdonald. It introduces the $\mathrm{Od}$-function and the workhorse formula that relate $\mathrm{Od}(f^{\lambda})$ to $\mathrm{Od}(f^{\mu})$ across $2^R$-parents, enabling recursive computation of $a_1(n)$ and $a_3(n)$ via the auxiliary function $\delta(n)=a_1(n)-a_3(n)$. The authors derive explicit formulas for sparse binary expansions and for the special case $n=2^{R}+2^{R-1}$, yielding closed forms and parity-based recursions such as $\delta(n)=(2-2(-1)^m)\delta(m)$ in the sparse regime. They also detail counting arguments for Type I and II $2^R$-parents and discuss open problems in the general, non-sparse setting, highlighting the role of $2^R$-cores and the limitations of current techniques. Overall, the work connects deep combinatorial machinery (hooks, cores, and $\beta$-sets) to the modular behavior of partition dimensions, with potential implications for understanding spinorial representations of symmetric groups.

Abstract

The number of standard Young tableaux of shape 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. 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.

Enumeration of Odd Dimensional Partitions modulo 4

TL;DR

This paper advances the enumeration of partitions by the dimension modulo 4, refining earlier results of McKay and Macdonald. It introduces the -function and the workhorse formula that relate to across -parents, enabling recursive computation of and via the auxiliary function . The authors derive explicit formulas for sparse binary expansions and for the special case , yielding closed forms and parity-based recursions such as in the sparse regime. They also detail counting arguments for Type I and II -parents and discuss open problems in the general, non-sparse setting, highlighting the role of -cores and the limitations of current techniques. Overall, the work connects deep combinatorial machinery (hooks, cores, and -sets) to the modular behavior of partition dimensions, with potential implications for understanding spinorial representations of symmetric groups.

Abstract

The number of standard Young tableaux of shape a partition is called the dimension of the partition and is denoted by . Partitions with odd dimensions were enumerated by McKay and were further characterized by Macdonald. Let be the number of partitions of with dimension congruent to modulo 4. In this paper, we refine Macdonald's and McKay's results by computing and when has no consecutive 1s in its binary expansion or when the sum of binary digits of is 2.
Paper Structure (24 sections, 24 theorems, 70 equations, 1 table)

This paper contains 24 sections, 24 theorems, 70 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. Structure of the paper
  5. Notations and Definitions
  6. Hooks and hook-lengths
  7. Hook removal and cores
  8. $\beta$-sets
  9. Odd dimensional partitions
  10. Od function
  11. The Workhorse Formula
  12. Odd part of factorial modulo 4
  13. The workhorse formula
  14. Understanding $\eta^\lambda_\mu$
  15. Odd Partitions of Sparse Numbers
  16. ...and 9 more sections

Key Result

Theorem 1

Let $n, m, R \in \mathbb{N}$ with $R\geq 2$ and $m > 0$. Suppose, $n = 2^{R} + m$ with $2^{R-1} > m$. Then we have

Theorems & Definitions (76)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Definition 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Example 9
  • Definition 10: $t$-core
  • ...and 66 more