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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)

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

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)

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)

In this work, we put to light a formula that relies the number of fully packed loop configurations (FPLs) associated to a given coupling π to the number of half-turn symmetric FPLs (HTFPLs) of even size whose coupling is a punctured version of the coupling π. When the coupling π is the coupling with all arches parallel π 0 (the " rarest " one), this formula… (More)

We introduce and study new combinatorial objects called Dyck tableaux which may be seen as a variant of permutation tableaux. These objects appear in the combinatorial interpretation of the physical model PASEP (Partially Simple Asymmetric Exclusion Process). Dyck tableaux afford a simple recursive structure through the construction of an insertion… (More)

The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn. The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestricted ASM's and the number of half-turn symmetric ASM's. Résumé. L'objet de ce travail est d'´ enumérer les matrices` a signes… (More)