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Congruences for hook lengths of partitions

Frédéric Jouhet, David Wahiche

TL;DR

The paper investigates congruences for hook-length statistics across partition families, emphasizing self-conjugate and $z$-asymmetric partitions. It leverages the Littlewood decomposition to derive addition and multiplication theorems, yielding explicit generating functions and congruences for the number of hooks of length $t$ in various partition classes, including $\,\mathcal{SC}$, $\,\mathcal{BG}_{z,t}$, and related sets. Key contributions include proving congruences for even $t$ in self-conjugate partitions, extending results to odd $t$ via restricted BG sets, and establishing modular Nekrasov–Okounkov-type identities for the broader BG$_{z,t}$ framework. These results unify and extend prior work (Han–Ji, Bessenrodt–Bacher–Manivel, AAOS, AK) and deepen the understanding of hook-length arithmetic across partition families with representation-theoretic connections.

Abstract

Recently, Amdeberhan et al. proved congruences for the number of hooks of fixed even length among the set of self-conjugate partitions of an integer $n$, therefore answering positively a conjecture raised by Ballantine et al.. In this paper, we show how these congruences can be immediately derived and generalized from an addition theorem for self-conjugate partitions proved by the second author. We also recall how the addition theorem proved before by Han and Ji can be used to derive similar congruences for the whole set of partitions, which are originally due to Bessenrodt, and Bacher and Manivel. Finally, we extend such congruences to the set of $z$-asymmetric partitions defined by Ayyer and Kumari, by proving an addition-multiplication theorem for these partitions. Among other things, this contains as special cases the congruences for the number of hook lengths for the self-conjugate and the so-called doubled distinct partitions.

Congruences for hook lengths of partitions

TL;DR

The paper investigates congruences for hook-length statistics across partition families, emphasizing self-conjugate and -asymmetric partitions. It leverages the Littlewood decomposition to derive addition and multiplication theorems, yielding explicit generating functions and congruences for the number of hooks of length in various partition classes, including , , and related sets. Key contributions include proving congruences for even in self-conjugate partitions, extending results to odd via restricted BG sets, and establishing modular Nekrasov–Okounkov-type identities for the broader BG framework. These results unify and extend prior work (Han–Ji, Bessenrodt–Bacher–Manivel, AAOS, AK) and deepen the understanding of hook-length arithmetic across partition families with representation-theoretic connections.

Abstract

Recently, Amdeberhan et al. proved congruences for the number of hooks of fixed even length among the set of self-conjugate partitions of an integer , therefore answering positively a conjecture raised by Ballantine et al.. In this paper, we show how these congruences can be immediately derived and generalized from an addition theorem for self-conjugate partitions proved by the second author. We also recall how the addition theorem proved before by Han and Ji can be used to derive similar congruences for the whole set of partitions, which are originally due to Bessenrodt, and Bacher and Manivel. Finally, we extend such congruences to the set of -asymmetric partitions defined by Ayyer and Kumari, by proving an addition-multiplication theorem for these partitions. Among other things, this contains as special cases the congruences for the number of hook lengths for the self-conjugate and the so-called doubled distinct partitions.

Paper Structure

This paper contains 5 sections, 14 theorems, 72 equations, 3 figures.

Table of Contents

  1. Introduction and notations
  2. Consequences of the addition theorem and a multiplication theorem for $\mathcal{SC}$
  3. Combinatorial properties of the Littlewood decomposition on $z$-asymmetric partitions
  4. Proof of the addition-multiplication theorem for $z$-asymmetric partitions
  5. Some consequences

Key Result

Theorem 1.1

HJ Let $t$ be a positive integer and set $\rho$ a function defined on $\mathbb{N}$. Let $g_t$ be the following formal power series: Then we have

Figures (3)

  • Figure 1: Ferrers diagram and some partition statistics
  • Figure 2: An example of doubled distinct and its conjugate
  • Figure 3: For $\lambda=(5,5,2,2)$, $s(\lambda)=\dots01100\mid 111001\dots$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 16 more