18 research outputs found
The signed Eulerian numbers on involutions
We define an analogue of signed Eulerian numbers for involutions of
the symmetric group and derive some combinatorial properties of this sequence.
In particular, we exhibit both an explicit formula and a recurrence for
arising from the properties of its generating function.Comment: 10 page
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
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
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
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