16 research outputs found
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
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
Equidistribution of negative statistics and quotients of Coxeter groups of type B and D
We generalize some identities and q-identities previously known for the
symmetric group to Coxeter groups of type B and D. The extended results include
theorems of Foata and Sch\"utzenberger, Gessel, and Roselle on various
distributions of inversion number, major index, and descent number. In order to
show our results we provide caracterizations of the systems of minimal coset
representatives of Coxeter groups of type B and D.Comment: 18 pages, 2 figure
Parabolically induced functions and equidistributed pairs
Given a function defined over a parabolic subgroup of a Coxeter group,
equidistributed with the length, we give a procedure to construct a function
over the entire group, equidistributed with the length. Such a procedure
permits to define functions equidistributed with the length in all the finite
Coxeter groups. We can establish our results in the general setting of graded
posets which satisfy some properties. These results apply to some known
functions arising in Coxeter groups as the major index, the negative major
index and the D-negative major index defined in type , and
respectively
Enumerating Sn by associated transpositions and linear extensions of finite posets
AbstractWe define a family of statistics over the symmetric group Sn indexed by subsets of the transpositions, and we study the corresponding generating functions. We show that they have many interesting combinatorial properties. In particular we prove that any poset of size n corresponds to a subset of transpositions of Sn, and that the generating function of the corresponding statistic includes partial linear extensions of such a poset. We prove equidistribution results, and we explicitly compute the associated generating functions for several classes of subsets