23 research outputs found
Signed Mahonians
A classical result of MacMahon gives a simple product formula for the
generating function of major index over the symmetric group. A similar
factorial-type product formula for the generating function of major index
together with sign was given by Gessel and Simion. Several extensions are given
in this paper, including a recurrence formula, a specialization at roots of
unity and type analogues.Comment: 23 page
Signed mahonians on some trees and parabolic quotients
We study the distribution of the major index with sign on some parabolic
quotients of the symmetric group, extending and generalizing simultaneously
results Gessel-Simion and Adin-Gessel-Roichman, and on some special trees that
we call rakes. We further consider and compute the distribution of the
flag-major index on some parabolic quotients of wreath products and other
related groups. All these distributions turn out to have very simple
factorization formulas.Comment: 12 page
Odd length for even hyperoctahedral groups and signed generating functions
We define a new statistic on the even hyperoctahedral groups which is a
natural analogue of the odd length statistic recently defined and studied on
Coxeter groups of types and . We compute the signed (by length)
generating function of this statistic over the whole group and over its maximal
and some other quotients and show that it always factors nicely. We also
present some conjectures
Equidistribution and Sign-Balance on 321-Avoiding Permutations
Let be the set of 321-avoiding permutations of order . Two
properties of are proved: (1) The {\em last descent} and {\em last index
minus one} statistics are equidistributed over , and also over subsets of
permutations whose inverse has an (almost) prescribed descent set. An analogous
result holds for Dyck paths. (2) The sign-and-last-descent enumerators for
and are essentially equal to the last-descent enumerator
for . The proofs use a recursion formula for an appropriate multivariate
generating function.Comment: 17 pages; to appear in S\'em. Lothar. Combi
Bimahonian distributions
Motivated by permutation statistics, we define for any complex reflection
group W a family of bivariate generating functions. They are defined either in
terms of Hilbert series for W-invariant polynomials when W acts diagonally on
two sets of variables, or equivalently, as sums involving the fake degrees of
irreducible representations for W. It is also shown that they satisfy a
``bicyclic sieving phenomenon'', which combinatorially interprets their values
when the two variables are set equal to certain roots of unity.Comment: Final version to appear in J. London Math. So
Counting derangements with signed right-to-left minima and excedances
Recently Alexandersson and Getachew proved some multivariate generalizations
of a formula for enumerating signed excedances in derangements. In this paper
we first relate their work to a recent continued fraction for permutations and
confirm some of their observations. Our second main result is two refinements
of their multivariate identities, which clearly explain the meaning of each
term in their main formulas.
We also explore some similar formulas for permutations of type B.Comment: Advances in Applied Mathematics 152, 10259