Arbres minimax et polynômes d'André

Abstract

On the set of minimax trees of a given order there can be defined two families of operations, the complements and the reverses. We study the actions of those operations and show that their orbits are enumerated by combinatorial objects previously introduced, such as the Hetyei-Reiner trees, the increasing trees, and the André trees. Various generating functions for those trees by several statistics are also derived.  2001 Elsevier Science

Counting Simsun Permutations by Descents

“…In [5], Foata and Han show that r(t)) and remark that (3.2) was first proved by Foata and Schützenberger [6]. In view of Proposition 3.2, we are able to obtain an equivalent algebraic expression for D(x, t), as follows.…”

“…We shall need in the present section the notions of minimax trees, increasing trees, and André trees, where the former notion is due to Foata and Han [5]. A labelled binary tree T is said to be minimax if in any subtree of T , the label of the root of the subtree is either the minimum, or the maximum, of the labels of vertices of the subtree.…”

“…It follows that André trees are increasing minimax trees such that for any internal vertex s, the right subtree T r (s) is nonempty (and necessarily contain the vertex of maximum label in T (s)). In [5], Foata and Han call minimax trees satisfying the André property "increasing Hetyei-Reiner trees," the latter class of trees was considered in the work of Hetyei and Reiner [9], which inspired the generalization due to Foata and Han.…”

“…In [5], Foata and Han describe a bijection sending increasing binary trees of order n to permutations in S n , which, when restricted to André trees, yields a bijection sending André trees of order n to "André permutations" of length n. On the other hand, according to the above definition of André trees, the rightmost leave of an André tree must have the largest label so that the corresponding "André permutation" π has its last letter the largest. Any permutation having its last letter the largest is said to be augmented.…”

“…Foata and Han [5] proved that D n (t) = ∑ k 0 d n, k t k , the generating function of André trees of order n by the number of leaves, or the generating function of augmented André permutations (see the discussion above) of length n by the number of peaks, satisfies the following recurrence relation [5, (7.1)]: Proof. Let d n (t) := t −1 D n+1 (t).…”