796 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
Actions on permutations and unimodality of descent polynomials
We study a group action on permutations due to Foata and Strehl and use it to
prove that the descent generating polynomial of certain sets of permutations
has a nonnegative expansion in the basis ,
. This property implies symmetry and unimodality. We
prove that the action is invariant under stack-sorting which strengthens recent
unimodality results of B\'ona. We prove that the generalized permutation
patterns and are invariant under the action and use this to
prove unimodality properties for a -analog of the Eulerian numbers recently
studied by Corteel, Postnikov, Steingr\'{\i}msson and Williams.
We also extend the action to linear extensions of sign-graded posets to give
a new proof of the unimodality of the -Eulerian polynomials of
sign-graded posets and a combinatorial interpretations (in terms of
Stembridge's peak polynomials) of the corresponding coefficients when expanded
in the above basis.
Finally, we prove that the statistic defined as the number of vertices of
even height in the unordered decreasing tree of a permutation has the same
distribution as the number of descents on any set of permutations invariant
under the action. When restricted to the set of stack-sortable permutations we
recover a result of Kreweras.Comment: 19 pages, revised version to appear in Europ. J. Combi
Mahonian STAT on words
In 2000, Babson and Steingr\'imsson introduced the notion of what is now
known as a permutation vincular pattern, and based on it they re-defined known
Mahonian statistics and introduced new ones, proving or conjecturing their
Mahonity. These conjectures were proved by Foata and Zeilberger in 2001, and by
Foata and Randrianarivony in 2006.
In 2010, Burstein refined some of these results by giving a bijection between
permutations with a fixed value for the major index and those with the same
value for STAT, where STAT is one of the statistics defined and proved to be
Mahonian in the 2000 Babson and Steingr\'imsson's paper. Several other
statistics are preserved as well by Burstein's bijection.
At the Formal Power Series and Algebraic Combinatorics Conference (FPSAC) in
2010, Burstein asked whether his bijection has other interesting properties. In
this paper, we not only show that Burstein's bijection preserves the Eulerian
statistic ides, but also use this fact, along with the bijection itself, to
prove Mahonity of the statistic STAT on words we introduce in this paper. The
words statistic STAT introduced by us here addresses a natural question on
existence of a Mahonian words analogue of STAT on permutations. While proving
Mahonity of our STAT on words, we prove a more general joint equidistribution
result involving two six-tuples of statistics on (dense) words, where
Burstein's bijection plays an important role
Counting Dyck paths by area and rank
The set of Dyck paths of length inherits a lattice structure from a
bijection with the set of noncrossing partitions with the usual partial order.
In this paper, we study the joint distribution of two statistics for Dyck
paths: \emph{area} (the area under the path) and \emph{rank} (the rank in the
lattice).
While area for Dyck paths has been studied, pairing it with this rank
function seems new, and we get an interesting -refinement of the Catalan
numbers. We present two decompositions of the corresponding generating
function: one refines an identity of Carlitz and Riordan; the other refines the
notion of -nonnegativity, and is based on a decomposition of the
lattice of noncrossing partitions due to Simion and Ullman.
Further, Biane's correspondence and a result of Stump allow us to conclude
that the joint distribution of area and rank for Dyck paths equals the joint
distribution of length and reflection length for the permutations lying below
the -cycle in the absolute order on the symmetric group.Comment: 24 pages, 7 figures. Connections with work of C. Stump
(arXiv:0808.2822v2) eliminated the need for 5 pages of proof in the first
draf
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