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Finite analogs of partition bias related to hook length two and a variant of Sylvester's map

Alexander Berkovich, Aritram Dhar

TL;DR

The paper develops finite analogs of a hook-length bias between partitions into odd parts and distinct parts by bounding the largest part. It derives closed $q$-series generating functions for the total number of hooks of length two, proves a finite bias via a Sylvester-map-based injection, and extends the results to weighted counts with $\binom{m}{2}$ hooks. By taking the bound to infinity, the authors obtain explicit nonnegative expressions for the difference in weighted counts, thereby establishing a bias in the unbounded case. These results contribute to the theory of partition inequalities and $q$-series by connecting Sylvester's map with finite analogs of hook-length distributions.

Abstract

In this paper, we count the total number of hooks of length two in all odd partitions of $n$ and all distinct partitions of $n$ with a bound on the largest part of the partitions. We generalize inequalities of Ballantine, Burson, Craig, Folsom and Wen by showing there is a bias in the number of hooks of length two in all odd partitions over all distinct partitions of $n$ in presence of a bound on the largest part. To establish such a bias, we use a variant of Sylvester's map. Then, we conjecture a similar finite bias for a weighted count of hooks of length two and prove it when we remove the bound on the largest part.

Finite analogs of partition bias related to hook length two and a variant of Sylvester's map

TL;DR

The paper develops finite analogs of a hook-length bias between partitions into odd parts and distinct parts by bounding the largest part. It derives closed -series generating functions for the total number of hooks of length two, proves a finite bias via a Sylvester-map-based injection, and extends the results to weighted counts with hooks. By taking the bound to infinity, the authors obtain explicit nonnegative expressions for the difference in weighted counts, thereby establishing a bias in the unbounded case. These results contribute to the theory of partition inequalities and -series by connecting Sylvester's map with finite analogs of hook-length distributions.

Abstract

In this paper, we count the total number of hooks of length two in all odd partitions of and all distinct partitions of with a bound on the largest part of the partitions. We generalize inequalities of Ballantine, Burson, Craig, Folsom and Wen by showing there is a bias in the number of hooks of length two in all odd partitions over all distinct partitions of in presence of a bound on the largest part. To establish such a bias, we use a variant of Sylvester's map. Then, we conjecture a similar finite bias for a weighted count of hooks of length two and prove it when we remove the bound on the largest part.

Paper Structure

This paper contains 12 sections, 18 theorems, 54 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proofs of Main Results
  4. Proof of Theorem \ref{['thm21']}
  5. Proof of Theorem \ref{['thm22']}
  6. Proof of Theorem \ref{['thm23']}
  7. Proof of Theorem \ref{['thm24']}
  8. Proof of Theorem \ref{['thm25']}
  9. Proof of Corollary \ref{['cor27']}
  10. Proof of Corollary \ref{['cor28']}
  11. Proof of Theorem \ref{['thm29']}
  12. Concluding Remarks

Key Result

Theorem 1.1

For any complex number $z$, we have

Figures (7)

  • Figure 1: The Young diagram of the partition $\pi = (5,3,2)$.
  • Figure 2: An example of Sylvester's bijection: $\pi_o = (7,5,3,3)$ (read row by row). $\pi_d = \psi(\pi_o) = (7,6,4,1)$. The red lines are the odd-indexed parts of $\pi_d$ and the green lines are the even-indexed parts of $\pi_d$. It is clear that $\gamma(\pi_d) = 4$.
  • Figure 3: The Young diagram of the partition $\pi = (8,6,5,2,1)$ with its hook lengths.
  • Figure 4: The distinct partitions $\pi_d$ of $n=7$ having $l(\pi_d)\le 5$ and their hook lengths.
  • Figure 5: $\varphi(((\color{blue}6\color{black},\color{blue}5\color{black}),\color{red}\emptyset\color{black})) = ((\color{blue}11\color{black}),\color{red}\emptyset\color{black})$
  • ...and 2 more figures

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Conjecture 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Example 1.9
  • Theorem 2.1
  • ...and 18 more