Related papers: Hook length biases and general linear partition in…
We establish a hook length bias between self-conjugate partitions and partitions of distinct odd parts, demonstrating that there are more hooks of fixed length $t \geq 2$ among self-conjugate partitions of $n$ than among partitions of…
In this article, we study hook lengths of ordinary partitions and $t$-regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in $2$-regular…
Recently, the theory of hook length biases has emerged as a prominent research topic. Led by Ballantine, Burson, Craig, Folsom, and Wen [\textit{Res. Math. Sci.}, 2023], hook length biases are being explored for ordinary partitions, odd…
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,…
Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular…
Recently, there has been a lot of work on combinatorial inequalities related to hook-lengths in $t$-regular partitions. In this short note, we give a proof using generating functions for a result proved by Singh and Barman (2026) using…
In this paper, we consider the asymptotic properties of hook numbers of partitions in restricted classes. More specifically, we compare the frequency with which partitions into odd parts and partitions into distinct parts have hook numbers…
In this article, we study hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions. More precisely, we establish hook length inequalities between $\ell$-regular partitions and $\ell$-distinct partitions for hook lengths $2$…
Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular…
Let $b_{n,k}$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. Singh and Barman raised the question of finding the relation between $b_{t,2}(n)$ and $b_{t,1}(n)$. Kim showed that there exists $N$ such that…
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…
The dimension of an irreducible representation of $GL(n,\mathbb{C})$, $Sp(2n)$, or $SO(n)$ is given by the respective hook-length and content formulas for the corresponding partition. The first author, inspired by the Nekrasov-Okounkov…
Recently, Amdeberhan et al. proved congruences for the number of hooks of fixed even length among the set of self-conjugate partitions of an integer $n$, therefore answering positively a conjecture raised by Ballantine et al.. In this…
Let $b_{t,i}(n)$ denote the total number of the $i$ hooks in the $t$-regular partitions of $n$. Singh and Barman (J. Number Theory { 264} (2024), 41--58) raised two conjectures on $b_{t,i}(n)$. The first conjecture is on the positivity of…
This paper shows that the number of hooks of length k contained in all partitions of n equals k times the number of parts of length k in all partitions of n. It contains also formulas for the moments (under uniform distribution) of k-th…
We confirm the speculation that the distribution of $t$-hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the number of hooks of length $t$ among the size $n$ self-conjugate…
Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub's analogue of Euler's…
Some conjectures on partition hook lengths, recently stated by the author, have been proved and generalized by Stanley, who also needed a formula by Andrews, Goulden and Jackson on symmetric functions to complete his derivation. Another…
In a recent paper, Bringmann, Craig, Ono, and the author showed that the number of $t$-hooks ($t\geq2$) among all partitions of $n$ is not always asymptotically equidistributed on congruence classes $a \pmod{b}$. In this short note, we…
Let $p_t(a,b;n)$ denote the number of partitions of $n$ such that the number of $t$ hooks is congruent to $a \bmod{b}$. For $t\in \{2, 3\}$, arithmetic progressions $r_1 \bmod{m_1}$ and $r_2 \bmod{m_2}$ on which $p_t(r_1,m_1; m_2 n + r_2)$…