Matthieu Josuat-Vergès

A q-enumeration of alternating permutations

A classical result of Euler states that the tangent numbers are an alternating sum of Eulerian numbers. A dual result of Roselle states that the secant numbers can be obtained by a signed enumeration of derangements. We show that both identities can be refined with the following statistics: the number of crossings in permutations and derangements, and the number of patterns 31-2 in alternating permutations. Using previous results of Corteel, Rubey, Prellberg, and the author, we derive closed formulas for both q-tangent and q-secant numbers. There are two different methods for obtaining these formulas: one with permutation tableaux and one with weighted Motzkin paths (Laguerre histories).

Crossings, Motzkin paths and moments

Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain q-analogues of Laguerre and Charlier polynomials. The moments of these orthogonal polynomials have combinatorial models in terms of crossings in permutations and set partitions. The aim of this article is to prove simple formulae for the moments of the q-Laguerre and the q-Charlier polynomials, in the style of the Touchard-Riordan formula (which gives the moments of some q-Hermite polynomials, and also the distribution of crossings in matchings). Our method mainly consists of the enumeration of weighted Motzkin paths, which are naturally associated with the moments. Some steps are bijective, in particular, we describe a decomposition of paths which generalises a previous construction of Penaud for the case of the Touchard-Riordan formula. There are also some non-bijective steps using basic hypergeometric series, and continued fractions or, alternatively, functional equations.

Bijections between pattern-avoiding fillings of Young diagrams

The pattern-avoiding fillings of Young diagrams we study arose from Postnikovs work on positive Grassmann cells. They are called -diagrams, and are in bijection with decorated permutations. Other closely-related fillings are interpreted as acyclic orientations of some bipartite graphs. The definition of the diagrams is the same but the avoided patterns are different. We give here bijections proving that the number of pattern-avoiding filling of a Young diagram is the same, for these two different sets of patterns. The result was obtained by Postnikov via a recurrence relation. This relation was extended by Spiridonov to obtain more general results about other patterns and other polyominoes than Young diagrams, and we show that our bijections also extend to more general polyominoes.