59 research outputs found
Symmetric unimodal expansions of excedances in colored permutations
We consider several generalizations of the classical -positivity of
Eulerian polynomials (and their derangement analogues) using generating
functions and combinatorial theory of continued fractions. For the symmetric
group, we prove an expansion formula for inversions and excedances as well as a
similar expansion for derangements. We also prove the -positivity for
Eulerian polynomials for derangements of type . More general expansion
formulae are also given for Eulerian polynomials for -colored derangements.
Our results answer and generalize several recent open problems in the
literature.Comment: 27 pages, 10 figure
Cyclic derangements
A classic problem in enumerative combinatorics is to count the number of
derangements, that is, permutations with no fixed point. Inspired by a recent
generalization to facet derangements of the hypercube by Gordon and McMahon, we
generalize this problem to enumerating derangements in the wreath product of
any finite cyclic group with the symmetric group. We also give q- and (q,
t)-analogs for cyclic derangements, generalizing results of Brenti and Gessel.Comment: 14 page
A symmetric unimodal decomposition of the derangement polynomial of type
The derangement polynomial for the symmetric group enumerates
derangements by the number of excedances. The derangement polynomial
for the hyperoctahedral group is a natural type analogue. A new
combinatorial formula for this polynomial is given in this paper. This formula
implies that decomposes as a sum of two nonnegative, symmetric and
unimodal polynomials whose centers of symmetry differ by a half and thus
provides a new transparent proof of its unimodality. A geometric
interpretation, analogous to Stanley's interpretation of as the local
-polynomial of the barycentric subdivision of the simplex, is given to one
of the summands of this decomposition. This interpretation leads to a unimodal
decomposition and a new formula for the Eulerian polynomial of type . The
various decomposing polynomials introduced here are also studied in terms of
recurrences, generating functions, combinatorial interpretations, expansions
and real-rootedness.Comment: Changes in Remark 7.3 and the bibliograph
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
The local -vector of the cluster subdivision of a simplex
The cluster complex is an abstract simplicial complex,
introduced by Fomin and Zelevinsky for a finite root system . The
positive part of naturally defines a simplicial subdivision of
the simplex on the vertex set of simple roots of . The local -vector
of this subdivision, in the sense of Stanley, is computed and the corresponding
-vector is shown to be nonnegative. Combinatorial interpretations to
the entries of the local -vector and the corresponding -vector are
provided for the classical root systems, in terms of noncrossing partitions of
types and . An analogous result is given for the barycentric subdivision
of a simplex.Comment: 21 pages, 4 figure
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