32,780 research outputs found
The sorting index
We consider a bivariate polynomial that generalizes both the length and
reflection length generating functions in a finite Coxeter group. In seeking a
combinatorial description of the coefficients, we are led to the study of a new
Mahonian statistic, which we call the sorting index. The sorting index of a
permutation and its type B and type D analogues have natural combinatorial
descriptions which we describe in detail.Comment: 14 pages, minor changes, new references adde
A two-sided analogue of the Coxeter complex
For any Coxeter system of rank , we introduce an abstract boolean
complex (simplicial poset) of dimension that contains the Coxeter
complex as a relative subcomplex. Faces are indexed by triples , where
and are subsets of the set of simple generators, and is a
minimal length representative for the parabolic double coset . There
is exactly one maximal face for each element of the group . The complex is
shellable and thin, which implies the complex is a sphere for the finite
Coxeter groups. In this case, a natural refinement of the -polynomial is
given by the "two-sided" -Eulerian polynomial, i.e., the generating function
for the joint distribution of left and right descents in .Comment: 26 pages, several large tables and figure
A note on three types of quasisymmetric functions
In the context of generating functions for -partitions, we revisit three
flavors of quasisymmetric functions: Gessel's quasisymmetric functions, Chow's
type B quasisymmetric functions, and Poirier's signed quasisymmetric functions.
In each case we use the inner coproduct to give a combinatorial description
(counting pairs of permutations) to the multiplication in: Solomon's type A
descent algebra, Solomon's type B descent algebra, and the Mantaci-Reutenauer
algebra, respectively. The presentation is brief and elementary, our main
results coming as consequences of -partition theorems already in the
literature.Comment: 10 page
The -colored composition poset
We generalize Bj\"{o}rner and Stanley's poset of compositions to -colored
compositions. Their work draws many analogies between their (1-colored)
composition poset and Young's lattice of partitions, including links to
(quasi-)symmetric functions and representation theory. Here we show that many
of these analogies hold for any number of colors. While many of the proofs for
Bj\"{o}rner and Stanley's poset were simplified by showing isomorphism with the
subword order, we remark that with 2 or more colors, our posets are not
isomorphic to a subword order.Comment: 12 pages, 1 figur
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