Flows on rooted trees and the Narayana idempotents

Several generating series for flows on rooted trees are introduced, as elements in the group of series associated with the Pre-Lie operad. By combinatorial arguments, one proves identities that characterise these series. One then gives a complete description of the image of these series in the group of series associated with the Dendriform operad. This allows to recover the Lie idempotents in the descent algebras recently introduced by Menous, Novelli and Thibon. Moreover, one defines new Lie idempotents and conjecture the existence of some others.Comment: 37 pages ; 6 figure

Ramanujan-Bernoulli numbers as moments of Racah polynomials

The classical sequence of Bernoulli numbers is known to the the sequence of moments of a family of orthogonal polynomials. Some similar statements are obtained for another sequence of rational numbers, which is similar in many ways to the Bernoulli numbers

On some varieties associated with trees

This article considers some affine algebraic varieties attached to finite trees and closely related to cluster algebras. Their definition involves a canonical coloring of vertices of trees into three colors. These varieties are proved to be smooth and to admit sometimes free actions of algebraic tori. Some results are obtained on their number of points over finite fields and on their cohomology.Comment: 37 pages, 7 figure

Fine structures inside the PreLie operad

This article aims at a detailed analysis of the PreLie operad. We obtain a more concrete description of the relationship between the anticyclic structure of PreLie and the generators of PreLie as a Lie-module, which was known before only at the level of characters. Building on this, we obtain an inclusion of the cyclic Lie module in the PreLie operad. We conjecture that the image of this inclusion generates an interesting free sub-operad.Comment: 16 page

On intervals in some posets of forests

A nice factorization is given for the characteristic polynomials of intervals in some posets of leaf-labeled forests of rooted binary trees.Comment: 14 pages, 5 figure