Some remarks on the joint distribution of descents and inverse descents
2013 (English)In: The Electronic Journal of Combinatorics, ISSN 1077-8926, Vol. 20, no 1, P52- p.Article in journal (Refereed) Published
We study the joint distribution of descents and inverse descents over the set of permutations of n letters. Gessel conjectured that the two-variable generating function of this distribution can be expanded in a given basis with nonnegative integer coefficients. We investigate the action of the Eulerian operators that give the recurrence for these generating functions. As a result we devise a recurrence for the coefficients in question but are unable to settle the conjecture. We examine generalizations of the conjecture and obtain a type B analog of the recurrence satisfied by the two-variable generating function. We also exhibit some connections to cyclic descents and cyclic inverse descents. Finally, we propose a combinatorial model for the joint distribution of descents and inverse descents in terms of statistics on inversion sequences.
Place, publisher, year, edition, pages
2013. Vol. 20, no 1, P52- p.
Permutations, descents, inverse descents, Eulerian numbers
IdentifiersURN: urn:nbn:se:kth:diva-120285ISI: 000315952700001ScopusID: 2-s2.0-84874890380OAI: oai:DiVA.org:kth-120285DiVA: diva2:614552
QC 201304052013-04-052013-04-042013-04-05Bibliographically approved