Combinatorics of the three-parameter PASEP partition function

We consider a partially asymmetric exclusion process (PASEP) on a finite number of sites with open and directed boundary conditions. Its partition function was calculated by Blythe, Evans, Colaiori, and Essler. It is known to be a generating function of permutation tableaux by the combinatorial interpretation of Corteel and Williams. We prove bijectively two new combinatorial interpretations. The first one is in terms of weighted Motzkin paths called Laguerre histories and is obtained by refining a bijection of Foata and Zeilberger. Secondly we show that this partition function is the generating function of permutations with respect to right-to-left minima, right-to-left maxima, ascents, and 31-2 patterns, by refining a bijection of Francon and Viennot. Then we give a new formula for the partition function which generalizes the one of Blythe & al. It is proved in two combinatorial ways. The first proof is an enumeration of lattice paths which are known to be a solution of the Matrix Ansatz of Derrida & al. The second proof relies on a previous enumeration of rook placements, which appear in the combinatorial interpretation of a related normal ordering problem. We also obtain a closed formula for the moments of Al-Salam-Chihara polynomials.Comment: 31 page

Generalized Dumont-Foata polynomials and alternative tableaux

Dumont and Foata introduced in 1976 a three-variable symmetric refinement of Genocchi numbers, which satisfies a simple recurrence relation. A six-variable generalization with many similar properties was later considered by Dumont. They generalize a lot of known integer sequences, and their ordinary generating function can be expanded as a Jacobi continued fraction. We give here a new combinatorial interpretation of the six-variable polynomials in terms of the alternative tableaux introduced by Viennot. A powerful tool to enumerate alternative tableaux is the so-called "matrix Ansatz", and using this we show that our combinatorial interpretation naturally leads to a new proof of the continued fraction expansion.Comment: 17 page

Stammering tableaux

The PASEP (Partially Asymmetric Simple Exclusion Process) is a probabilistic model of moving particles, which is of great interest in combinatorics, since it appeared that its partition function counts some tableaux. These tableaux have several variants such as permutations tableaux, alternative tableaux, tree- like tableaux, Dyck tableaux, etc. We introduce in this context certain excursions in Young's lattice, that we call stammering tableaux (by analogy with oscillating tableaux, vacillating tableaux, hesitating tableaux). Some natural bijections make a link with rook placements in a double staircase, chains of Dyck paths obtained by successive addition of ribbons, Laguerre histories, Dyck tableaux, etc.Comment: Clarification and better exposition thanks reviewer's report

Refined enumeration of noncrossing chains and hook formulas

In the combinatorics of finite finite Coxeter groups, there is a simple formula giving the number of maximal chains of noncrossing partitions. It is a reinterpretation of a result by Deligne which is due to Chapoton, and the goal of this article is to refine the formula. First, we prove a one-parameter generalization, by the considering enumeration of noncrossing chains where we put a weight on some relations. Second, we consider an equivalence relation on noncrossing chains coming from the natural action of the group on set partitions, and we show that each equivalence class has a simple generating function. Using this we recover Postnikov's hook length formula in type A and obtain a variant in type B.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:1304.090

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

És sostenible el nostre sistema de pensions?

En el document es discuteix l'afirmació estàndard sobre que el sistema de pensions de jubilació públiques a Espanya -i, per extensió, els sistemes públics europeus- no és financerament viable. Es mostra perquè la pròpia interrogació retòrica que recull el títol és de fet una pregunta mal formulada (oblida coses essencials). I es posa de manifest que un sistema de pensions de jubilació privat -com els dels fons de pensions de les entitats financeres- sols és viable per a una minoria de la població: la de rendes més elevade