Related papers: An involution on set partitions
Statistics of distinguishable particles has become relevant in systems of colloidal particles and in the context of applications of statistical mechanics to complex networks. When studying these type of systems with the standard textbook…
In this paper, we introduce a natural geometric extension of the partition function. More precisely, we investigate the problem of counting partitions of a rectangle into rectangular blocks with integer sides. Here, two partitions of a…
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…
Interesting properties of the partitions of a finite field $\mathbb F_q$ induced by the combination of involutions and trace maps are studied. The special features of involutions of the form $\frac{u}{z}$, $u$ being a fixed element of…
We show that many theorems which assert that two kinds of partitions of the same integer $n$ are equinumerous are actually special cases of a much stronger form of equality. We show that in fact there correspond partition statistics $X$ and…
We study the enumeration of set partitions, according to their length, number of parts, cyclic type, and genus. We introduce genus-dependent Bell, Stirling numbers, and Fa\`a di Bruno coefficients. Besides attempting to summarize what is…
The fragmentation processes of exchangeable partitions have already been studied by several authors. In this paper, we examine rather fragmentation of exchangeable compositions, that means partitions of $\mathbb{N}$ where the order of the…
I present a bijection on integer partitions that leads to recursive expressions, closed formulae and generating functions for the cardinality of certain sets of partitions of a positive integer $n$. The bijection leads also to a product on…
A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work we prove a strengthening of this theorem originally conjectured by Haglund. Our…
We prove a lemma that is useful to get upper bounds for the number of partitions without a given subsum. From this we can deduce an improved upper bound for the number of sets represented by the (unrestricted or into unequal parts)…
Recently, many authors have investigated how various partition statistics distribute as the size of the partition grows. In this work, we look at a particular statistic arising from the recent rejuvenation of MacMahon's partition analysis.…
This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A…
We prove a conjecture of Haglund which can be seen as an extension of the equidistribution of the inversion number and the major index over permutations to ordered set partitions. Haglund's conjecture implicitly defines two statistics on…
In [Decompositions and statistics for \beta(1,0)-trees and nonseparable permutations, Advances Appl. Math. 42 (2009) 313--328] we introduced an involution, h, on \beta(1,0)-trees. We neglected, however, to prove that h indeed is an…
Given a sequence $S=(s_1,\dots,s_m) \in [0, 1]^m$, a block $B$ of $S$ is a subsequence $B=(s_i,s_{i+1},\dots,s_j)$. The size $b$ of a block $B$ is the sum of its elements. It is proved in [1] that for each positive integer $n$, there is a…
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.
A conjecture by Deutsch, Kitaev, and Remmel states that the triples of permutation statistics $(S_{10}, S_{12}, S_{17})$ and $(S_{12}, S_{10} ,S_{17})$ are equidistributed over the symmetric group $\mathfrak{S}_n$. Here, $S_{10}$ enumerates…
We introduce and study a new notion of patterns in Stirling and $k$-Stirling permutations, which we call block patterns. We prove a general result which allows us to compute generating functions for the occurrences of various block patterns…
Starting from a master equation, we derive the evolution equation for the size distribution of elements in an evolving system, where each element can grow, divide into two, and produce new elements. We then probe general solutions of the…
Let $(X_1,X_2,...)$ be a random partition of the unit interval $[0,1]$, i.e. $X_i\geq0$ and $\sum_{i\geq1} X_i=1$, and let $(\varepsilon_1,\varepsilon_2,...)$ be i.i.d. Bernoulli random variables of parameter $p \in (0,1)$. The Bernoulli…