Jean-Christophe Aval

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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)