4,723 research outputs found
On some generalized -Eulerian polynomials
The -Eulerian polynomials are the enumerative polynomials of permutations. Using Shareshian and Wachs' exponential generating function of these Eulerian polynomials, Chung and Graham proved two symmetrical -Eulerian identities and asked for bijective proofs. We provide such proofs using Foata and Han's three-variable statistic . We also prove a new recurrence formula for the -Eulerian polynomials and study a -analogue of Chung and Graham's restricted Eulerian polynomials. In particular, we obtain a symmetrical identity for these restricted -Eulerian polynomials with a combinatorial proof
The Interlace Polynomial
In this paper, we survey results regarding the interlace polynomial of a
graph, connections to such graph polynomials as the Martin and Tutte
polynomials, and generalizations to the realms of isotropic systems and
delta-matroids.Comment: 18 pages, 5 figures, to appear as a chapter in: Graph Polynomials,
edited by M. Dehmer et al., CRC Press/Taylor & Francis Group, LL
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.
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