8 research outputs found
Solving multivariate functional equations
This paper presents a new method to solve functional equations of
multivariate generating functions, such as
giving a
formula for in terms of a sum over finite sequences. We use this
method to show how one would calculate the coefficients of the generating
function for parallelogram polyominoes, which is impractical using other
methods. We also apply this method to answer a question from fully commutative
affine permutations.Comment: 11 pages, 1 figure. v3: Main theorems and writing style revised for
greater clarity. Updated to final version, to appear in Discrete Mathematic
Permutations with forbidden subsequences and a generalized Schröder number
AbstractUsing the technique of generating trees, we prove that there are exactly 10 classes of pattern avoiding permutations enumerated by the large Schröder numbers. For each integer, m⩾1, a sequence which generalizes the Schröder and Catalan numbers is shown to enumerate m+22 classes of pattern avoiding permutations. Combinatorial interpretations in terms of binary trees and polyominoes and a generating function for these sequences are given
Combinatorics of fully commutative involutions in classical Coxeter groups
An element of a Coxeter group is fully commutative if any two of its
reduced decompositions are related by a series of transpositions of adjacent
commuting generators. In the present work, we focus on fully commutative
involutions, which are characterized in terms of Viennot's heaps. By encoding
the latter by Dyck-type lattice walks, we enumerate fully commutative
involutions according to their length, for all classical finite and affine
Coxeter groups. In the finite cases, we also find explicit expressions for
their generating functions with respect to the major index. Finally in affine
type , we connect our results to Fan--Green's cell structure of the
corresponding Temperley--Lieb algebra.Comment: 25 page
The enumeration of fully commutative affine permutations
We give a generating function for the fully commutative affine permutations
enumerated by rank and Coxeter length, extending formulas due to Stembridge and
Barcucci--Del Lungo--Pergola--Pinzani. For fixed rank, the length generating
functions have coefficients that are periodic with period dividing the rank. In
the course of proving these formulas, we obtain results that elucidate the
structure of the fully commutative affine permutations.Comment: 18 pages; final versio
The excedances and descents of bi-increasing permutations
Starting from some considerations we make about the relations between certain
difference statistics and the classical permutation statistics we study
permutations whose inversion number and excedance difference coincide. It turns
out that these (so-called bi-increasing) permutations are just the 321-avoiding
ones. The paper investigates their excedance and descent structure. In
particular, we find some nice combinatorial interpretations for the
distribution coefficients of the number of excedances and descents,
respectively, and their difference analogues over the bi-increasing
permutations in terms of parallelogram polyominoes and 2-Motzkin paths. This
yields a connection between restricted permutations, parallelogram polyominoes,
and lattice paths that reveals the relations between several well-known
bijections given for these objects (e.g. by Delest-Viennot,
Billey-Jockusch-Stanley, Francon-Viennot, and Foata-Zeilberger). As an
application, we enumerate skew diagrams according to their rank and give a
simple combinatorial proof for a result concerning the symmetry of the joint
distribution of the number of excedances and inversions, respectively, over the
symmetric group.Comment: 36 page