123 research outputs found
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
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
Permutation patterns and statistics
Let S_n denote the symmetric group of all permutations of the set {1, 2,
...,n} and let S = \cup_{n\ge0} S_n. If Pi is a set of permutations, then we
let Av_n(Pi) be the set of permutations in S_n which avoid every permutation of
Pi in the sense of pattern avoidance. One of the celebrated notions in pattern
theory is that of Wilf-equivalence, where Pi and Pi' are Wilf equivalent if
#Av_n(Pi)=#Av_n(Pi') for all n\ge0. In a recent paper, Sagan and Savage
proposed studying a q-analogue of this concept defined as follows. Suppose
st:S->N is a permutation statistic where N represents the nonnegative integers.
Consider the corresponding generating function, F_n^{st}(Pi;q) = sum_{sigma in
Av_n(Pi)} q^{st sigma}, and call Pi,Pi' st-Wilf equivalent if
F_n^{st}(Pi;q)=F_n^{st}(Pi';q) for all n\ge0. We present the first in-depth
study of this concept for the inv and maj statistics. In particular, we
determine all inv- and maj-Wilf equivalences for any Pi containd in S_3. This
leads us to consider various q-analogues of the Catalan numbers, Fibonacci
numbers, triangular numbers, and powers of two. Our proof techniques use
lattice paths, integer partitions, and Foata's fundamental bijection. We also
answer a question about Mahonian pairs raised in the Sagan-Savage article.Comment: 28 pages, 5 figures, tightened up the exposition, noted that some of
the conjectures have been prove
A central limit theorem for descents of a Mallows permutation and its inverse
This paper studies the asymptotic distribution of descents \des(w) in a
permutation , and its inverse, distributed according to the Mallows measure.
The Mallows measure is a non-uniform probability measure on permutations
introduced to study ranked data. Under this measure, permutations are weighted
according to the number of inversions they contain, with the weighting
controlled by a parameter . The main results are a Berry-Esseen theorem for
\des(w)+\des(w^{-1}) as well as a joint central limit theorem for
(\des(w),\des(w^{-1})) to a bivariate normal with a non-trivial correlation
depending on . The proof uses Stein's method with size-bias coupling along
with a regenerative process associated to the Mallows measure.Comment: v2 some added references and minor changes to introduction. 35 pages,
1 figure, 1 table. Comments are welcome
A Combinatorial Formula for Macdonald Polynomials
We prove a combinatorial formula for the Macdonald polynomial H_mu(x;q,t)
which had been conjectured by the first author. Corollaries to our main theorem
include the expansion of H_mu(x;q,t) in terms of LLT polynomials, a new proof
of the charge formula of Lascoux and Schutzenberger for Hall-Littlewood
polynomials, a new proof of Knop and Sahi's combinatorial formula for Jack
polynomials as well as a lifting of their formula to integral form Macdonald
polynomials, and a new combinatorial rule for the Kostka-Macdonald coefficients
K_{lambda,mu}(q,t) in the case that mu is a partition with parts less than or
equal to 2.Comment: 29 page
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