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

Cohomology rings of toric varieties assigned to cluster quivers: the case of unioriented quivers of type A

The theory of cluster algebras of S. Fomin and A. Zelevinsky has assigned a fan to each Dynkin diagram. Then A. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov have generalized this construction using arbitrary quivers on Dynkin diagrams. In the special case of the unioriented quiver of type A, we describe the cohomology ring of the toric variety associated to this fan. A natural base is obtained and an explicit rule is given for the product of any two generators.Comment: 12 page