Jean-Christophe Aval

Learn More
The purpose of this paper is to find a new way to prove the n! conjecture for particular partitions. The idea is to construct a monomial and explicit basis for the space M µ. We succeed completely for hook-shaped partitions, i.e., µ = (K +1, 1 L). We are able to exhibit a basis and to verify that its cardinality is indeed n!, that it is linearly independent(More)
In this work we introduce and study tree-like tableaux, which are certain fillings of Ferrers diagrams in simple bijection with permutation tableaux and alternative tableaux. We exhibit an elementary insertion procedure on our tableaux which gives a clear proof that tree-like tableaux of size n are counted by n!, and which moreover respects most of the(More)
Catalan numbers C(n) = 1 n+1 2n n enumerate binary trees and Dyck paths. The distribution of paths with respect to their number k of factors is given by ballot numbers B(n, k) = n−k n+k n+k n. These integers are known to satisfy simple recurrence, which may be visualised in a " Catalan triangle " , a lower-triangular two-dimensional array. It is surprising(More)
The aim of this work is to study some lattice diagram determinants ∆L(X, Y) as defined in [5] and to extend results of [3]. We recall that ML denotes the space of all partial derivatives of ∆L. In this paper, we want to study the space M k i,j (X, Y) which is defined as the sum of ML spaces where the lattice diagrams L are obtained by removing k cells from(More)
This article investigates combinatorial properties of non-ambiguous trees. These objects we define may be seen either as binary trees drawn on a grid with some constraints, or as a subset of the tree-like tableaux previously defined by Aval, Boussicault and Nadeau. The enumeration of non-ambiguous trees satisfying some additional constraints allows us to(More)
We obtain explicit formulas for the enumeration of labelled parallelogram polyominoes. These are the polyominoes that are bounded, above and below, by northeast lattice paths going from the origin to a point (k, n). The numbers from 1 to n (the labels) are bijectively attached to the n north steps of the above-bounding path, with the condition that they(More)