Related papers: Statistics of Partial Permutations via Catalan mat…
We introduce new natural generalizations of the classical descent and inversion statistics for permutations, called width-$k$ descents and width-$k$ inversions. These variations induce generalizations of the excedance and major statistics,…
We propose sum rules for permutations $p_n(k)$ of the ensemble $\left\{1,2,\cdots,n\right\}$ with $k$ fixed points, in the form of partial sums of their moments. The corresponding identities involve Stirling numbers of the first kind…
Writing the values of Krawtchouk polynomials as matrices, we consider weighted partial sums along columns. For the general case, we find an identity that, in the symmetric case yields a formula for such partial sums. Complete sums of…
The Super-Catalan numbers are a generalization of the Catalan numbers defined as $T(m,n) = \frac{(2m)!(2n)!}{2m!n!(m+n)!}$. It is an open problem to find a combinatorial interpretation for $T(m,n)$. We resolve this for $m=3,4$ using a…
We present some results on the proportion of permutations of length $n$ containing certain mesh patterns as $n$ grows large, and give exact enumeration results in some cases. In particular, we focus on mesh patterns where entire rows and…
Permutation tableaux are combinatorial objects related with permutations and various statistics on them. They appeared in connection with total positivity in Grassmannians, and stationary probabilities in a PASEP model. In particular they…
Curious spectral properties of an ensemble of random unitary matrices appearing in the quantization of a map p -> p+alpha, q -> q+f(p+alpha) in [Giraud et al. nlin.CD/0403033] are investigated. When alpha=m/n with integer co-prime m,n and…
Let $T$ be a tree on $n$ vertices. We can regard the edges of $T$ as transpositions of the vertex set; their product (in any order) is a cyclic permutation. All possible cyclic permutations arise (each exactly once) if and only if the tree…
During the study of dual sequences, Sun introduced the polynomials \[ D_n(x,y)=\sum_{k=0}^{n}{n\choose k}{x\choose k}y^k\text{ and } S_n(x,y)=\sum_{k=0}^{n}\binom{n}{k}\binom{x}{k}\binom{-1-x}{k} y^k. \] Many related congruences have been…
The link between a particular class of growth processes and random matrices was established in the now famous 1999 article of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation. During the…
By a very simple argument, we prove that if $l,m,n$ are nonnegative integers then $$\sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m} =\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}. On the basis of this…
Recently, a new class of words, denoted by L_n, was shown to be in bijection with a subset of the Dyck paths of length 2n having cardinality given by the (n-1)-st Catalan number. Here, we consider statistics on L_n recording the number of…
In studying the enumerative theory of super characters' of the group of upper triangular matrices over a finite field we found that the moments (mean, variance and higher moments) of novel statistics on set partitions have simple closed…
Let $[n]$ be a finite chain $\{1, 2, \ldots, n\}$, and let $\mathcal{IC}_{n}$ be the semigroup consisting of all isotone and order-decreasing injective partial transformations on $[n]$. In addition, let $\mathcal{Q}^{\prime}_{n} = \{\alpha…
We introduce a statistic $\pmaj$ on partitions of $[n]=\{1,2,..., n\}$, and show that it is equidistributed with the number of 2-crossings over partitions of $[n]$ with given sets of minimal block elements and maximal block elements. This…
Let $X_i,i=0,1,\ldots$ be a sequence of iid random variables whose distribution is continuous. Associated with this sequence is the sequence $(i,X_i),i=0,1,\ldots$. Let ${\cal R}_{n}$ denote the set of Pareto optimal elements of $\{…
To flatten a set partition (with apologies to Mathematica) means to form a permutation by erasing the dividers between its blocks. Of course, the result depends on how the blocks are listed. For the usual listing--increasing entries in each…
We use the concept of the half of a lower-triangular matrix to define a transformation on integer sequences. We explore the properties of this transformation, including in some cases a study of the Hankel transform of the transformed…
In this (partly expository) paper we give a short overview about the close relationship between the sequence of Catalan numbers and Hankel determinants from the point of view of orthogonal polynomials and show that an analogous situation…
We prove that the inverse of the Hankel matrix of the reciprocals of the Catalan numbers has integer entries. We generalize the result to an infinite family of generalized Catalan numbers. The Hankel matrices that we consider are associated…