相关论文: Congruences for hook lengths of partitions

Recently Andrews and Bachraoui proved identities relating certain restricted partitions into distinct even parts with restricted 4-regular partitions by the theory of basic hypergeometric series. They also posed a question regarding…

In 2009, the first author proved the Nekrasov-Okounkov formula on hook lengths for integer partitions by using an identity of Macdonald in the framework of type $\widetilde A$ affine root systems, and conjectured that some summations over…

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…

In a paper by the author, Hemmer, Hopkins, and Keith the concept of a fixed point in a sequence was applied to the sequence of first column hook lengths of a partition. In this paper we generalize this notion to fixed hook lengths in an…

In a recent paper, Bacher and de la Harpe study conjugacy growth series of infinite permutation groups and their relationships with $p(n)$, the partition function, and $p(n)_{\textbf{e}}$, a generalized partition function. They prove…

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, 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…

Strongly refining results by Regev, Regev and Zeilberger, we prove surprising coincidences between characters to 2-part partitions of size n and characters to hooks of size n+2 on two related families obtained by extending 2-regular…

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…

In this paper, we generalize recent work of Mizuhara, Sellers, and Swisher that gives a method for establishing restricted plane partition congruences based on a bounded number of calculations. Using periodicity for partition functions, our…

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…

The combinatorial properties of partitions with various restrictions on their hooksets are explored. A connection with numerical semigroups extends current results on simultaneous s/t-cores. Conditions that suffice for a partition to…

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…

We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part…

We obtain congruences for the number a(n) of cubic partitions using modular forms. The notion of cubic partitions is introduced by Chan and named by Kim in connection with Ramanujan's cubic continued fractions. Chan has shown that a(n) has…

Using only a symmetric p-core partition and p-quotient, we give an explicit formula for the set of diagonal hook lengths of the associated symmetric partition.

Simultaneous core partitions have been widely studied since Anderson's work on the enumeration of $(s,t)$-core partitions. Amdeberhan and Leven showed that the number of $(s,s+1, \ldots, s+k)$-core partitions is equal to the number of $(s,…

In this paper we present an extension of Stanley's theorem related to partitions of positive integers. Stanley's theorem states a relation between "the sum of the numbers of distinct members in the partitions of a positive integer $n$" and…

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…

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…