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Sum of squares of hook lengths and contents

Krishna Menon

TL;DR

This paper proves a bijective identity linking sums of hook lengths and contents for any partition $\lambda \vdash n$ via a constructive map $H(\lambda) \to N(\lambda) \cup C(\lambda)$, thereby establishing $\sum_{u\in\lambda} h(u)^2 = n^2 + \sum_{u\in\lambda} c(u)^2$. It then derives a rectangle-counting formula $|\text{Rect}(\lambda)| = \sum_{u\in\lambda} h(u) + \tfrac{1}{2}\sum_{u\in\lambda} \tilde{h}(u)^2$, where $\tilde{h}(u)$ is the partial hook length, and interprets thick rectangles via a map in the construction. The authors extend the approach to sums of higher powers, proving $\sum h(u)^k \le n^k + \sum |c(u)|^k$ for $k\ge 3$ with equality iff $\lambda$ is a hook, and outline generalized sets $H^k,N^k,C^k$ with corresponding maps; the $k=1$ case yields a similar inequality with a hook-characterization. Overall, the work provides a bijective understanding of hook-length/content relations and a combinatorial framework for rectangle counting and higher-power identities in partitions.

Abstract

It is known that for the Young diagram of any partition of an integer $n$, the sum of squares of the hook lengths of its cells is exactly $n^2$ more than that of the contents of its cells. That is, for any partition $λ$ of an integer $n$, \begin{equation*} \sum_{u \in λ} h(u)^2 = n^2 + \sum_{u \in λ} c(u)^2. \end{equation*} We provide a bijective proof of this fact, thus solving a problem posed by Stanley. Along the way, we obtain a formula for the number of rectangles in the Young diagram of a partition. We also mention a result for sums of other powers of hook lengths and contents.

Sum of squares of hook lengths and contents

TL;DR

This paper proves a bijective identity linking sums of hook lengths and contents for any partition via a constructive map , thereby establishing . It then derives a rectangle-counting formula , where is the partial hook length, and interprets thick rectangles via a map in the construction. The authors extend the approach to sums of higher powers, proving for with equality iff is a hook, and outline generalized sets with corresponding maps; the case yields a similar inequality with a hook-characterization. Overall, the work provides a bijective understanding of hook-length/content relations and a combinatorial framework for rectangle counting and higher-power identities in partitions.

Abstract

It is known that for the Young diagram of any partition of an integer , the sum of squares of the hook lengths of its cells is exactly more than that of the contents of its cells. That is, for any partition of an integer , \begin{equation*} \sum_{u \in λ} h(u)^2 = n^2 + \sum_{u \in λ} c(u)^2. \end{equation*} We provide a bijective proof of this fact, thus solving a problem posed by Stanley. Along the way, we obtain a formula for the number of rectangles in the Young diagram of a partition. We also mention a result for sums of other powers of hook lengths and contents.
Paper Structure (5 sections, 6 theorems, 6 equations, 7 figures)

This paper contains 5 sections, 6 theorems, 6 equations, 7 figures.

Table of Contents

  1. Preliminaries
  2. Bijective proof of \ref{['thm']}
  3. Rectangles in partitions
  4. Other powers
  5. Acknowledgements

Key Result

Theorem 1.1

For any $n \geq 1$ and $\lambda \vdash n$, we have

Figures (7)

  • Figure 1: Young diagram of $(5, 5, 4, 2)$.
  • Figure 2: Highlighted cells form the hook of the starred cell.
  • Figure 3: Elements of $H(\lambda), N(\lambda), C(\lambda)$ from \ref{['objex']}.
  • Figure 4: Elements of $N_1(\lambda)$ associated to certain elements of $H_1(\lambda)$ under $\phi_1$.
  • Figure 5: Methods for relabeling an element of $H_2(\lambda)$.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Theorem 1.1
  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof : Proof of \ref{['thm']}
  • Remark 2.5
  • ...and 5 more