Related papers: Hook length biases and general linear partition in…

Partitions, the partition function $p(n)$, and the hook lengths of their Ferrers-Young diagrams are important objects in combinatorics, number theory and representation theory. For positive integers $n$ and $t$, we study $p_t^e(n)$ (resp.…

We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions: \frac{1}{n!} \sum_{\lambda \vdash n} f_{\lambda}^2 \sum_{u \in \lambda} \prod_{i=1}^{r}(h_u^2 - i^2) = \frac{1}{2(r+1)^2}…

Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of $t$-hooks in the partitions of $n$. We prove that the limiting distribution is normal with mean $\mu_t(n)\sim…

Let $b_{t,i}(n)$ denote the total number of $i$-hooks in $t$-regular partitions of $n$. Singh and Barman conjectured that $b_{t+1,2}(n) \geq b_{t,2}(n)$ holds for all $t\ge 3$ and $n\ge 0$. This conjecture was known to hold for $t=3$ due to…

The Nekrasov--Okounkov hook length formula provides a fundamental link between the theory of partitions and the coefficients of powers of the Dedekind eta function. In this paper we examine three conjectures presented by Amdeberhan. The…

In this paper, we show that the difference between the number of parts in the odd partitions of $n$ and the number of parts in the distinct partitions of $n$ satisfies Euler's recurrence relation for the partition function $p(n)$ when $n$…

We use the hook lengths of a partition to define two rectangular tableaux. We prove these tableaux have equal multisets of entries, first by elementary combinatorial arguments, and then using Stanley's Hook Content Formula and symmetric…

We give relations between the joint distributions of multiple hook lengths and of frequencies and part sizes in partitions, extending prior work in this area. These results are discovered by investigating truncations of the…

We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the hook length and content statistics. As an application, several new hook-content formulas for…

Recently, Griffin, Ono, and Tsai examined the distribution of the number of $t$-hooks in partitions of $n$, which was later followed by the work of Craig, Ono, and Singh on the distribution of the number of $t$-hooks in self-conjugate…

We prove that the number $q(n)$ of partitions into distinct parts is log-concave for $n \geq 33$ and satisfies the higher order Tur\'an inequalities for $n\geq 121$ conjectured by Craig and Pun. In doing so, we establish explicit error…

Recently, Blecher and Knopfmacher applied the notion of fixed points to integer partitions. This has already been generalized and refined in various ways such as $h$-fixed points for an integer parameter $h$ by Hopkins and Sellers. Here, we…

The symplectic/orthogonal contents of partitions are related to the dimensions of irreducible representations of symplectic/orthogonal groups. In 2012, motivated by Nekrasov--Okounkov's hook-length formula and Stanley's hook-content…

Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he…

Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory, and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when…

A {\em pointed partition} of $n$ is a pair $(\lambda, v)$ where $\lambda\vdash n$ and $v$ is a cell in its Ferrers diagram. We construct an involution on pointed partitions of $n$ exchanging "hook length" and "part length". This gives a…

The number of sequences of odd length in strict partitions (denoted as $\mathrm{sol}$), which plays a pivotal role in the first paper of this series, is investigated in different contexts, both new and old. Namely, we first note a direct…

Let $\lambda$ be a partition, viewed as a Young diagram. We define the hook difference of a cell of $\lambda$ to be the difference of its leg and arm lengths. Define $h_{1,1}(\lambda)$ to be the number of cells of $\lambda$ with hook…

We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to set partitions that are matchings, we obtain three new…

This paper proves a generalization of a conjecture of Guoniu Han, inspired originally by an identity of Nekrasov and Okounkov. The main result states that certain sums over partitions p of n, involving symmetric functions of the squares of…