1,068 research outputs found
width-k Eulerian polynomials of type A and B and its Gamma-positivity
We define some generalizations of the classical descent and inversion
statistics on signed permutations that arise from the work of Sack and
Ulfarsson [20] and called after width-k descents and width-k inversionsof type
A in Davis's work [8]. Using the aforementioned new statistics, we derive some
new generalizations of Eulerian polynomials of type A, B and D. It should also
be noticed that we establish the Gamma-positivity of the "width-k" Eulerian
polynomials and we give a combinatorial interpretation of finite sequences
associated to these new polynomials using quasisymmetric functions and
P-partition in Petersen's work [18].Comment: 28 page
Eulerian quasisymmetric functions
We introduce a family of quasisymmetric functions called {\em Eulerian
quasisymmetric functions}, which specialize to enumerators for the joint
distribution of the permutation statistics, major index and excedance number on
permutations of fixed cycle type. This family is analogous to a family of
quasisymmetric functions that Gessel and Reutenauer used to study the joint
distribution of major index and descent number on permutations of fixed cycle
type. Our central result is a formula for the generating function for the
Eulerian quasisymmetric functions, which specializes to a new and surprising
-analog of a classical formula of Euler for the exponential generating
function of the Eulerian polynomials. This -analog computes the joint
distribution of excedance number and major index, the only of the four
important Euler-Mahonian distributions that had not yet been computed. Our
study of the Eulerian quasisymmetric functions also yields results that include
the descent statistic and refine results of Gessel and Reutenauer. We also
obtain -analogs, -analogs and quasisymmetric function analogs of
classical results on the symmetry and unimodality of the Eulerian polynomials.
Our Eulerian quasisymmetric functions refine symmetric functions that have
occurred in various representation theoretic and enumerative contexts including
MacMahon's study of multiset derangements, work of Procesi and Stanley on toric
varieties of Coxeter complexes, Stanley's work on chromatic symmetric
functions, and the work of the authors on the homology of a certain poset
introduced by Bj\"orner and Welker.Comment: Final version; to appear in Advances in Mathematics; 52 pages; this
paper was originally part of the longer paper arXiv:0805.2416v1, which has
been split into three paper
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.
The symmetric and unimodal expansion of Eulerian polynomials via continued fractions
This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J.
Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the
coefficients in an expansion of generalized Eulerian polynomials, which implies
the symmetric and unimodal property of the Eulerian numbers. We show that such
a formula with the conjectured property can be derived from the combinatorial
theory of continued fractions. We also discuss an analogous expansion for the
corresponding formula for derangements and prove a -analogue of the fact
that the (-1)-evaluation of the enumerator polynomials of permutations (resp.
derangements) by the number of excedances gives rise to tangent numbers (resp.
secant numbers). The -analogue unifies and generalizes our recent
results (European J. Combin. \textbf{31} (2010), no.~7, 1689--1705.) and that
of Josuat-Verg\`es (European J. Combin. \textbf{31} (2010), no.~7, 1892--1906).Comment: 19 pages, 2 figure
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