کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4655447 | 1343384 | 2013 | 17 صفحه PDF | دانلود رایگان |
We prove a multivariate strengthening of Brentiʼs 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 Brentiʼs 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.
Journal: Journal of Combinatorial Theory, Series A - Volume 120, Issue 7, September 2013, Pages 1929–1945