224 research outputs found
A unified approach to polynomial sequences with only real zeros
We give new sufficient conditions for a sequence of polynomials to have only
real zeros based on the method of interlacing zeros. As applications we derive
several well-known facts, including the reality of zeros of orthogonal
polynomials, matching polynomials, Narayana polynomials and Eulerian
polynomials. We also settle certain conjectures of Stahl on genus polynomials
by proving them for certain classes of graphs, while showing that they are
false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres
Stable multivariate -Eulerian polynomials
We prove a multivariate strengthening of Brenti's result that every root of
the Eulerian polynomial of type 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 and . Finally,
although we are not able to settle Brenti's real-rootedness conjecture for
Eulerian polynomials of type , nor prove a companion conjecture of Dilks,
Petersen, and Stembridge for affine Eulerian polynomials of types and ,
we indicate some methods of attack and pose some related open problems.Comment: 17 pages. To appear in J. Combin. Theory Ser.
Recurrences for Eulerian polynomials of type B and type D
We introduce new recurrences for the type B and type D Eulerian polynomials,
and interpret them combinatorially. These recurrences are analogous to a
well-known recurrence for the type A Eulerian polynomials. We also discuss
their relationship to polynomials introduced by Savage and Visontai in
connection to the real-rootedness of the corresponding Eulerian polynomials
Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups
We use the method of mutual interlacing to prove two conjectures on the
real-rootedness of Eulerian-like polynomials: Brenti's conjecture on
-Eulerian polynomials for Weyl groups of type , and Dilks, Petersen, and
Stembridge's conjecture on affine Eulerian polynomials for irreducible finite
Weyl groups.
For the former, we obtain a refinement of Brenti's -Eulerian polynomials
of type , and then show that these refined Eulerian polynomials satisfy
certain recurrence relation. By using the Routh--Hurwitz theory and the
recurrence relation, we prove that these polynomials form a mutually
interlacing sequence for any positive , and hence prove Brenti's conjecture.
For , our result reduces to the real-rootedness of the Eulerian
polynomials of type , which were originally conjectured by Brenti and
recently proved by Savage and Visontai.
For the latter, we introduce a family of polynomials based on Savage and
Visontai's refinement of Eulerian polynomials of type . We show that these
new polynomials satisfy the same recurrence relation as Savage and Visontai's
refined Eulerian polynomials. As a result, we get the real-rootedness of the
affine Eulerian polynomials of type . Combining the previous results for
other types, we completely prove Dilks, Petersen, and Stembridge's conjecture,
which states that, for every irreducible finite Weyl group, the affine descent
polynomial has only real zeros.Comment: 28 page
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