Stable multivariate W-Eulerian polynomials
2013 (English)In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 120, no 7, 1929-1945 p.Article in journal (Refereed) Published
We prove a multivariate strengthening of Brenti's result that every root of the Eulerian polynomial of type B is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability a generalization of real-rootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator. Our results extend naturally to colored permutations, and we also give stable generalizations of recent real-rootedness results due to Dilks, Petersen, and Stembridge on affine Eulerian polynomials of types A and C. Finally, although we are not able to settle Brenti's real-rootedness conjecture for Eulerian polynomials of type D. nor prove a companion conjecture of Dilks, Petersen, and Stembridge for affine Eulerian polynomials of types B and D. we indicate some methods of attack and pose some related open problems.
Place, publisher, year, edition, pages
2013. Vol. 120, no 7, 1929-1945 p.
Eulerian polynomials, Descent set, Coxeter groups, Differential recurrence, Real roots only, Real stability, Catalan numbers
IdentifiersURN: urn:nbn:se:kth:diva-129447DOI: 10.1016/j.jcta.2013.07.009ISI: 000323868200037OAI: oai:DiVA.org:kth-129447DiVA: diva2:652864
QC 201310022013-10-022013-09-302013-10-02Bibliographically approved