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arXiv · 1203.0791

Stable multivariate $W$-Eulerian polynomials

Mirkó Visontai·Nathan Williams
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Abstract

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.

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BibTeXRIS

Mirkó Visontai, Nathan Williams. 2012-03-05. Stable multivariate $W$-Eulerian polynomials. https://doi.org/10.1016/j.jcta.2013.07.009

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