Tree-like Tableaux

Abstract

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 well-known statistics studied originally on alternative and permutation tableaux. Our insertion procedure allows to define in particular two simple new bijections between tree-like tableaux and permutations: the first one is conceived specifically to respect the generalized pattern 2-31, while the second one respects the underlying tree of a tree-like tableau.All authors are supported by the ANR (PSYCO project -ANR-11-JS02-001).

Probabilistic consequences of some polynomial recurrences

“…Our interest is motivated by the fact that polynomials satisfying such recurrences frequently appear as generating polynomials of integer valued random variables that are of interest in discrete mathematics. As an illustration, we will analyze probabilistic properties of tree-like tableaux, combinatorial objects that have recently been introduced by Aval, Boussicault, and Nadeau [2]. Since their introduction, they have been of interest to several mathematicians and are the topic of the papers [10,13,16,23].…”

“…Since their introduction, they have been of interest to several mathematicians and are the topic of the papers [10,13,16,23]. In [2], the expected number of diagonal cells in symmetric tree-like tableaux was computed and this paper extends this result by determining the variance and deriving the asymptotic distribution. The distribution is proven to be asymptotically normal and the proof of this reduces to the classical problem of determining whether a polynomial has real roots.…”

“…In this paper, we are interested in the subset consisting of symmetric tree-like tableaux which are tree-like tableaux symmetric about their main diagonal (see [2, Section 2.2] for more details and Figure 1(ii) for an example). As noticed in [2], the size of a symmetric tree-like tableaux must be odd, and it is known that there are n! tree-like tableaux of size n (see [2,Corollary 8]) and 2 n n!…”

“…As noticed in [2], the size of a symmetric tree-like tableaux must be odd, and it is known that there are n! tree-like tableaux of size n (see [2,Corollary 8]) and 2 n n! symmetric tree-like tableaux of size 2n + 1 (see [2,Corollary 8]).…”

“…tree-like tableaux of size n (see [2,Corollary 8]) and 2 n n! symmetric tree-like tableaux of size 2n + 1 (see [2,Corollary 8]).…”