303 research outputs found
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
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
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
Stokes posets and serpent nests
We study two different objects attached to an arbitrary quadrangulation of a
regular polygon. The first one is a poset, closely related to the Stokes
polytopes introduced by Baryshnikov. The second one is a set of some paths
configurations inside the quadrangulation, satisfying some specific
constraints. These objects provide a generalisation of the existing
combinatorics of cluster algebras and nonnesting partitions of type A.Comment: 24 pages, 12 figure
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