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Inequalities for the number of $t$-hooks in two partition classes arising from sum-product identities

Aritram Dhar, Byungchan Kim, Eunmi Kim, Ae Ja Yee

Abstract

Motivated by recent study on the number of $t$-hooks in partitions arising from Euler's partition identity, we investigate the number of $t$-hooks in the sets from the first Rogers-Ramanujan identity and the first little Göllitz identity. In particular, for $t=1,2$, we obtain the generating functions for the number of $t$-hooks and prove $t$-hook inequalities by deriving asymptotic formulas.

Inequalities for the number of $t$-hooks in two partition classes arising from sum-product identities

Abstract

Motivated by recent study on the number of -hooks in partitions arising from Euler's partition identity, we investigate the number of -hooks in the sets from the first Rogers-Ramanujan identity and the first little Göllitz identity. In particular, for , we obtain the generating functions for the number of -hooks and prove -hook inequalities by deriving asymptotic formulas.
Paper Structure (19 sections, 28 theorems, 187 equations, 1 figure)

This paper contains 19 sections, 28 theorems, 187 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $t$-hooks in two partition sets from the first Rogers--Ramanujan identity
  4. $t=1$ case
  5. $t=2$ case
  6. Asymptotics for hook numbers arising from the first Rogers--Ramanujan identity
  7. Asymptotics for $\mathcal{S}_{1,1}(q)$ in a narrow range of $y$
  8. Error estimate far from the peak
  9. Asymptotic of $S_{1,1}(q)$
  10. Proof of Theorem \ref{['thm:r1_asymp']}
  11. Proof of Theorem \ref{['thm:r2_asymp']}
  12. $t$-hooks in two partition sets from the first little Göllnitz Identity
  13. $t=1$ case
  14. $t=2$ case
  15. Asymptotics for hook numbers arising from Little Göllnitz identity
  16. ...and 4 more sections

Key Result

Theorem 1.2

For sufficiently large $n$,

Figures (1)

  • Figure 1: The Young diagram of the partition $\lambda = (7,4,2,2,1)$ with its hook lengths.

Theorems & Definitions (44)

  • Conjecture 1.1: Ballantine--Burson--Craig--Folsom--Wen ballantine1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1: BMRS
  • Lemma 2.2: BMRS
  • Proposition 2.3: BMRS
  • ...and 34 more