Olivier Guibert

A combinatorial proof of J. West's conjecture

Abstract We give what we believe to be the first combinatorial proof of J. Wests conjecture that is to say we show that two-stack sortable permutations are in correspondence with rooted nonseparable planar maps. The generating tree of a set of permutations with forbidden subsequences and its characterization by a rewriting system are used to obtain this correspondence. Other results on the number of two-stack sortable permutations with given classical parameters are presented.

Vexillary Involutions are Enumerated by Motzkin Numbers

Abstract. Vexillary permutations are very important for Schubert Polynomials. In this paper, we consider the enumeration of vexillary involutions, that is, 2143-avoiding involutions. Instead of solving the generating function obtained by a succession system characterizing vexillary involutions, we establish a one-to-one correspondence with 1-2 trees enumerated by Motzkin numbers.

Stack words, standard tableaux and Baxter permutations

Abstract The origin of this work is based on the enumeration of stack sortable permutations [11, 17, 18]. The problem, particularly in case of two stacks, exhibits classical objects in combinatorics such as permutations with forbidden subsequences, nonseparable planar maps [4, 5], and also standard Young tableaux if we are interested in the movements of stacks. So, we show that the number of 3 × n rectangular standard Young tableaux which avoid two consecutive integers on second row is c2n (where cn = (2n)!/(n + 1)!n!) and there is a one-to-one correspondence between the same tableaux which avoid two consecutive integers on the same row and Baxter permutations which are enumerated by . We also give formulas enumerating these objects according to various parameters.

Doubly alternating Baxter permutations are Catalan

The Baxter permutations who are alternating and whose inverse is also alternating are shown to be enumerated by the Catalan numbers. A bijection to complete binary trees is also given.

Enumeration of vexillary involutions which are equal to their mirror/complement

Abstract Vexillary permutations are very important for the study of Schubert polynomials which are involved in several areas of mathematics and physics. In this note we determine the number of n-length vexillary involutions, that is 2143-avoiding involutions, which are equal to their mirror/complement by establishing a bijection with n-length left factors of Motzkin words.

Enumeration of Some Labelled Trees

In this paper1 we are interested in the enumeration of rooted labelled trees according to the relationship between the root and its sons. Let T n,k be the family of Cayley trees on [n] such that the root has exactly k smaller sons. In a first time we give a bijective proof of |T n+1,K | = ( k n )n n−k . Moreover, we use the family T n+1,0 to give combinatorial explanations of various identities involving n n . We relate this family to the enumeration of minimal factorization of the n-cycle (1, 2, ..., n) as a product of transpositions. Finally, we use the fact that |T n+1,0| = n n to prove bijectively that there are 2n n ordered alternating trees on [n+1].

Stack words, standard Young tableaux, permutations with forbidden subsequences and planar maps

Abstract Stack words stem from studies on stack-sortable permutations and represent classical combinatorial objects such as standard Young tableaux, permutations with forbidden sequences and planar maps. We extend existing enumerative results on stack words and we also obtain new results. In particular, we make a correspondence between nonseparable 3× n rectangular standard Young tableaux (or stack words where elements satisfy a ‘Towers of Hanoi’ condition) and nonseparable cubic rooted planar maps with 2 n vertices enumerated by 2 n (3 n )!/((2 n +1)!( n +1)!). Moreover, these tableaux without two consecutive integers in the same row are in bijection with nonseparable rooted planar maps with n +1 edges enumerated by 2(3 n )!/((2 n +1)!( n +1)!).