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Hook lengths in self-conjugate partitions

Tewodros Amdeberhan, George E. Andrews, Ken Ono, Ajit Singh

Abstract

In 2010, G.-N. Han obtained the generating function for the number of size $t$ hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitions, which are particularly elegant for even $t$. If $n_t(λ)$ is the number of size $t$ hooks in a partition $λ,$ then for even $t$ we have $$\sum_{λ\in \mathcal{SC}} x^{n_t(λ)} q^{\vertλ\vert} = (-q;q^2)_{\infty} \cdot ((1-x^2)q^{2t};q^{2t})_{\infty}^{\frac{t}2}. $$ As a consequence, if $a_t^*(n)$ is the number of such hooks among the self-conjugate partitions of $n,$ then for even $t$ we obtain the simple formula $$ a_t^*(n)=t\sum_{j\geq 1} q^*(n-2tj), $$ where $q^*(m)$ is the number of partitions of $m$ into distinct odd parts. As a corollary, we find that $t\mid a_t^*(n),$ which confirms a conjecture of Ballantine, Burson, Craig, Folsom, and Wen.

Hook lengths in self-conjugate partitions

Abstract

In 2010, G.-N. Han obtained the generating function for the number of size hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitions, which are particularly elegant for even . If is the number of size hooks in a partition then for even we have As a consequence, if is the number of such hooks among the self-conjugate partitions of then for even we obtain the simple formula where is the number of partitions of into distinct odd parts. As a corollary, we find that which confirms a conjecture of Ballantine, Burson, Craig, Folsom, and Wen.
Paper Structure (6 sections, 7 theorems, 45 equations, 4 figures, 2 tables)

This paper contains 6 sections, 7 theorems, 45 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction and Statement of Results
  2. The refined Littlewood decomposition
  3. Proofs of Theorem 1.1 and Theorem 1.2.
  4. Counting 1-hooks in self-conjugate partitions
  5. Proof of Theorem \ref{['vital']}
  6. Proof of Theorem \ref{['ExactFormula']}

Key Result

Theorem 1.1

The following are true. (1) If $t$ is even, then we have (2) If $t$ is odd, then we have where $H^{\star}(x;q)$ is defined in (HstarFormula).

Figures (4)

  • Figure 1: Hook lengths of the self-conjugate partitions of $16$
  • Figure 2: $\lambda=(7,7,5,4,3,2,2)\in\mathcal{SC}$ with hook lengths inserted
  • Figure 3: The $4$-core $\omega$ and partitions $\nu^{(0)}, \nu^{(1)} , \nu^{(2)}, \nu^{(3)}$ for $\lambda$
  • Figure 4: Self-conjugate partitions of $17$ of Type 1 and Type 2

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Example
  • Conjecture : BBCFW
  • Corollary 1.3
  • Theorem 2.1
  • Example
  • Theorem 3.1
  • Example
  • ...and 5 more