50 research outputs found
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 Eulerian distribution on the involutions of the hyperoctahedral group is unimodal
The Eulerian distribution on the involutions of the symmetric group is
unimodal, as shown by Guo and Zeng. In this paper we prove that the Eulerian
distribution on the involutions of the hyperoctahedral group, when viewed as a
colored permutation group, is unimodal in a similar way and we compute its
generating function, using signed quasisymmetric functions.Comment: 11 pages, zero figure
Binomial Eulerian polynomials for colored permutations
Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and
Williams on the face enumeration of generalized permutohedra. They are
-positive (in particular, palindromic and unimodal) polynomials which
can be interpreted as -polynomials of certain flag simplicial polytopes and
which admit interesting Schur -positive symmetric function
generalizations. This paper introduces analogues of these polynomials for
-colored permutations with similar properties and uncovers some new
instances of equivariant -positivity in geometric combinatorics.Comment: Final version; minor change
-partitions and -positivity
Using the combinatorics of -unimodal sets, we establish two new
results in the theory of quasisymmetric functions. First, we obtain the
expansion of the fundamental basis into quasisymmetric power sums. Secondly, we
prove that generating functions of reverse -partitions expand positively
into quasisymmetric power sums. Consequently any nonnegative linear combination
of such functions is -positive whenever it is symmetric. As an application
we derive positivity results for chromatic quasisymmetric functions,
unicellular and vertical strip LLT polynomials, multivariate Tutte polynomials
and the more general -polynomials, matroid quasisymmetric functions, and
certain Eulerian quasisymmetric functions, thus reproving and improving on
numerous results in the literature.Comment: 47 pages, 4 figure