Related papers: Partitions with prescribed hooksets

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…

New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.

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…

Over the past twenty years, lecture hall partitions have emerged as fundamental combinatorial structures, leading to new generalizations and interpretations of classical theorems and new results. In recent years, geometric approaches to…

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 $p_t(r_1,m_1; m_2 n + r_2)$…

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…

Set partitions are arrangements of distinct objects into groups. The problem of listing all set partitions arises in a variety of settings, in particular in combinatorial optimization tasks. After a brief review, we give practical…

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.

Mass partition problems describe the partitions we can induce on a family of measures or finite sets of points in Euclidean spaces by dividing the ambient space into pieces. In this survey we describe recent progress in the area in addition…

We explore partitions that lie in the intersection of several sets of classical interest: partitions with parts indivisible by $m$, appearing fewer than $m$ times, or differing by less than $m$. We find results on their behavior and…

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…

For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.

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 introduces almost partitionable sets to generalize the known concept of partitionable sets. These notions provide a unified frame to construct $\mathbb{Z}$-cyclic patterned starter whist tournaments and cyclic balanced sampling…

We introduce a new concept, which we call partition expanders. The basic idea is to study quantitative properties of graphs in a slightly different way than it is in the standard definition of expanders. While in the definition of expanders…

The lattice of partitions of a set and its d-divisible generalization have been much studied for their combinatorial, topological, and representation-theoretic properties. An ordered set partition is a set partition where the subsets are…

Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpoint, and connect them with signed partitions and bicolored…

A construction is given of Markov partitions for some rational maps, which persist over regions of parameter space, not confined to single hyperbolic components. The set on which the Markov partition exists, and its boundary, are analysed.

Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split…

Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting…