Related papers: A generalization of conjugation of integer partiti…

This work highlights the existence of partial symmetries in large families of iterated plethystic coefficients. The plethystic coefficients involved come from the expansion in the Schur basis of iterated plethysms of Schur functions indexed…

Graphical models represent multivariate and generally not normalized probability distributions. Computing the normalization factor, called the partition function, is the main inference challenge relevant to multiple statistical and…

Recently, Schneider and Schneider defined a new class of partitions called sequentially congruent partitions, in which each part is congruent to the next part modulo its index, and they proved two partition bijections involving these…

Amdeberhan conjectured that the number of $(s,s+2)$-core partitions with distinct parts for an odd integer $s$ is $2^{s-1}$. This conjecture was first proved by Yan, Qin, Jin and Zhou, then subsequently by Zaleski and Zeilberger. Since the…

Let $\mathscr{S}$ denote the set of integer partitions into parts that differ by at least $3$, with the added constraint that no two consecutive multiples of $3$ occur as parts. We derive trivariate generating functions of Andrews--Gordon…

The number of parts in the partitions (resp. distinct partitions) of $n$ with parts from a set were considered. Its generating functions were obtained. Consequently, we derive several recurrence identities for the following functions: the…

We study statistics on ordered set partitions whose generating functions are related to $p,q$-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of \stein…

Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum…

We show that, for any fixed genus $g$, the ordinary generating function for the genus $g$ partitions of an $n$-element set into $k$ blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a…

In this article, we discuss the notion of partition of elements in an arbitrary Coxeter system $(W,S)$: a partition of an element $w$ is a subset $\mathcal P\subseteq W$ such that the left inversion set of $w$ is the disjoint union of the…

We introduce and study multiple partition structures which are sequences of probability measures on families of Young diagrams subjected to a consistency condition. The multiple partition structures are generalizations of Kingman's…

We introduce several statistics on ordered partitions of sets, that is, set partitions where the blocks are permuted arbitrarily. The distribution of these statistics is closely related to the q-Stirling numbers of the second kind. Some of…

This article is an extensive study of partitions with fixed number of odd and even-indexed odd parts. We use these partitions to generalize recent results of C. Savage and A. Sills. Moreover, we derive explicit formulas for generating…

We study certain bijection between plane partitions and $\mathbb{N}$-matrices. As applications, we prove a Cauchy-type identity for generalized dual Grothendieck polynomials. We introduce two statistics on plane partitions, whose generating…

The partition perimeter is a statistic defined to be one less than the sum of the number of parts and the largest part. Recently, Amdeberhan, Andrews, and Ballantine proved the following analog of Glaisher's theorem: for all $m \geq 2$ and…

We show how Andrews' generating functions for generalized Frobenius partitions can be understood within the theory of Eichler and Zagier as specific coefficients of certain Jacobi forms. This reformulation leads to a recursive process which…

Gibbs partitions of the integers generated by stable subordinators of index $\alpha\in(0,1)$ form remarkable classes of random partitions where in principle much is known about their properties, including practically effortless obtainment…

Using tools from representation theory, we derive expressions for the coincidence rate of partially-distinguishable particles in an interferometry experiment. Our expressions are valid for either bosons or fermions, and for any number of…

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 study the involution of the real line induced by the outer automorphism of the extended modular group PGL(2,Z). This `modular' involution is discontinuous at rationals but satisfies a surprising collection of functional equations. It…