Abstract
We give a simple bijection between some staircase tableaux and tables of inversion. Some nice properties of the bijection allows us to define some q-Eulerian polynomials related to the staircase tableaux. We also give a combinatorial interpretation of these q-Eulerian polynomials in terms of permutations. Nous proposons une bijection simple entre certains tableaux escalier et les tables d'inversion. Cette bijection nous permet de montrer que les statistiques Euleriennes et Mahoniennes sont naturelles sur les tableaux escalier. Nous définissons des polynÎmes q-Eulériens et en donnons une interprétation combinatoire.
Highlights
Staircase tableaux are new combinatorial objects defined by S
They are related to the asymmetric exclusion process on a one-dimensional lattice with open boundaries (ASEP) and were used to give a combinatorial formula for the moments of the Askey-Wilson polynomials defined in [1; 11]
We show that our very simple bijection can be generalized to any family of staircase tableaux
Summary
Staircase tableaux are new combinatorial objects defined by S. They are related to the asymmetric exclusion process on a one-dimensional lattice with open boundaries (ASEP) and were used to give a combinatorial formula for the moments of the Askey-Wilson polynomials defined in [1; 11]. We will show that the bijection allows us in this case to understand the statistic ânumber of ÎČs on the diagonalâ which is known to be related to the eulerian numbers [16; 18] Thanks to this we will introduce some new q-Eulerian polynomials and will give some combinatorial interpretation in terms of permutations. We end this extended abstract with some concluding remarks and open problems
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