40 research outputs found
Bijections for Permutation Tableaux
In this paper we propose a new bijection between permutation tableaux and permutations. This bijection shows how natural statistics on the tableaux are equidistributed to classical statistics on permutations: descents, RL-minima and pattern enumerations. We then use the bijection, and a related encoding of tableaux by words, to prove results about the enumeration of permutations with a fixed number of 31-2 patterns, and to define subclasses of permutation tableaux that are in bijection with set partitions. An extended version of this work is available in [6]
Generalized permutation patterns - a short survey
An occurrence of a classical pattern p in a permutation Ļ is a subsequence of Ļ whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidanceāor the prescribed number of occurrencesā of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns
The Asymptotic Distribution of Symbols on Diagonals of Random Weighted Staircase Tableaux
Staircase tableaux are combinatorial objects that were first introduced due
to a connection with the asymmetric simple exclusion process (ASEP) and
Askey-Wilson polynomials. Since their introduction, staircase tableaux have
been the object of study in many recent papers. Relevant to this paper, the
distri- bution of parameters on the first diagonal was proven to be
asymptotically normal. In that same paper, a conjecture was made that the other
diagonals would be asymptotically Poisson. Since then, only the second and the
third diagonal were proven to follow the conjecture. This paper builds upon
those results to prove the conjecture for fixed k. In particular, we prove that
the distribution of the number of alphas (betas) on the kth diagonal, k > 1, is
asymptotically Poisson with parameter 1\2. In addition, we prove that symbols
on the kth diagonal are asymptotically independent and thus, collectively
follow the Poisson distribution with parameter 1
The Matrix Ansatz, Orthogonal Polynomials, and Permutations
In this paper we outline a Matrix Ansatz approach to some problems of
combinatorial enumeration. The idea is that many interesting quantities can be
expressed in terms of products of matrices, where the matrices obey certain
relations. We illustrate this approach with applications to moments of
orthogonal polynomials, permutations, signed permutations, and tableaux.Comment: to appear in Advances in Applied Mathematics, special issue for
Dennis Stanto
Distribution of the Number of Corners in Tree-like and Permutation Tableaux
In this abstract, we study tree-like tableaux and some of their probabilistic properties. Tree-like tableaux are in bijection with other combinatorial structures, including permutation tableaux, and have a connection to the partially asymmetric simple exclusion process (PASEP), an important model of interacting particles system. In particular, in the context of tree-like tableaux, a corner corresponds to a node occupied by a particle that could jump to the right while inner corners indicate a particle with an empty node to its left. Thus, the total number of corners represents the number of nodes at which PASEP can move, i.e., the total current activity of the system. As the number of inner corners and regular corners is connected, we limit our discussion to just regular corners and show that, asymptotically, the number of corners in a tableaux of length n is normally distributed. Furthermore, since the number of corners in tree-like tableaux are closely related to the number of corners in permutation tableaux, we will discuss the corners in the context of the latter tableaux