Yvan Le Borgne

The sand-pile model and Tutte polynomials

A new explicit bijection between spanning trees and recurrent configurations of the sand-pile model is given. This mapping is such that the difference between the number of gains on a configuration and the external activity of the associate tree is the number of edges of the graph. It gives a bijective proof of a result of Merino Lopez expressing the generating function of recurrent configurations as an evaluation of the Tutte polynomial.

Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers

We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area,bounce) and (area,dinv) give rise to the same q,t-analogue of Narayana numbers which was introduced by two of the authors in [4]. We prove the main conjectures of that paper: the q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q,t-Narayana polynomials which relates them to the famous diagonal harmonics.