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We make a first step towards categorification of the dendriform operad, using categories of modules over the Tamari lattices. This means that we describe some functors that correspond to part of the operad structure.

- Jean-Christophe Aval
- Discrete Mathematics
- 2000

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)

- Jean-Christophe Aval, Adrien Boussicault, Philippe Nadeau
- Electr. J. Comb.
- 2013

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)

- Jean-Christophe Aval
- Discrete Mathematics
- 2008

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)

- Jean-Christophe Aval, Nantel Bergeron
- J. Comb. Theory, Ser. A
- 2002

A lattice diagram is a finite set L = {(p

- Jean-Christophe Aval
- Discrete Mathematics
- 2002

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 consider the partition function Z) of the square ice model with domain wall boundary. We give a simple proof of the symmetry of Z with respect to all its variables when the global parameter a of the model is set to the special value a = exp(iπ/3). Our proof does not use any determinantal interpretation of Z and can be adapted to other situations (for… (More)

Staircase tableaux are combinatorial objects which appear as key tools in the study of the PASEP physical model. The aim of this work is to show how the discovery of a tree structure in staircase tableaux is a significant feature to derive properties on these objects.

In [12], Lascoux and Schützenberger introduced a notion of key associated to any Young tableau. More recently Las-coux defined the key of an alternating sign matrix by recursively removing all −1's in such matrices. But alternating sign matrices are in bijection with monotone triangles, which form a subclass of Young tableaux. We show that in this case… (More)