32 research outputs found
Multichains, non-crossing partitions and trees
AbstractBijections are presented between certain classes of trees and multichains in non-crossing partition lattices
Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups
This memoir constitutes the author's PhD thesis at Cornell University. It
serves both as an expository work and as a description of new research. At the
heart of the memoir, we introduce and study a poset for each
finite Coxeter group and for each positive integer . When , our
definition coincides with the generalized noncrossing partitions introduced by
Brady-Watt and Bessis. When is the symmetric group, we obtain the poset of
classical -divisible noncrossing partitions, first studied by Edelman.
Along the way, we include a comprehensive introduction to related background
material. Before defining our generalization , we develop from
scratch the theory of algebraic noncrossing partitions . This involves
studying a finite Coxeter group with respect to its generating set of
{\em all} reflections, instead of the usual Coxeter generating set . This is
the first time that this material has appeared in one place.
Finally, it turns out that our poset shares many enumerative
features in common with the ``generalized nonnesting partitions'' of
Athanasiadis and the ``generalized cluster complexes'' of Fomin and Reading. In
particular, there is a generalized ``Fuss-Catalan number'', with a nice closed
formula in terms of the invariant degrees of , that plays an important role
in each case. We give a basic introduction to these topics, and we describe
several conjectures relating these three families of ``Fuss-Catalan objects''.Comment: Final version -- to appear in Memoirs of the American Mathematical
Society. Many small improvements in exposition, especially in Sections 2.2,
4.1 and 5.2.1. Section 5.1.5 deleted. New references to recent wor
Annular noncrossing permutations and minimal transitive factorizations
We give two combinatorial proofs of Goulden and Jackson's formula for the
number of minimal transitive factorizations of a permutation when the
permutation has two cycles. We use the recent result of Goulden, Nica, and
Oancea on the number of maximal chains of annular noncrossing partitions of
type .Comment: 13 pages, 3 Figure
An instance of umbral methods in representation theory: the parking function module
We test the umbral methods introduced by Rota and Taylor within the theory of
representation of symmetric group. We define a simple bijection between the set
of all parking functions of length and the set of all noncrossing
partitions of . Then we give an umbral expression of the
Frobenius characteristic of the parking function module introduced by Haiman
that allows an explicit relation between this symmetric function and the volume
polynomial of Pitman and Stanley
A self-dual poset on objects counted by the Catalan numbers and a type-B analogue
We introduce two partially ordered sets, and , of the same
cardinalities as the type-A and type-B noncrossing partition lattices. The
ground sets of and are subsets of the symmetric and the
hyperoctahedral groups, consisting of permutations which avoid certain
patterns. The order relation is given by (strict) containment of the descent
sets. In each case, by means of an explicit order-preserving bijection, we show
that the poset of restricted permutations is an extension of the refinement
order on noncrossing partitions. Several structural properties of these
permutation posets follow, including self-duality and the strong Sperner
property. We also discuss posets and similarly associated with
noncrossing partitions, defined by means of the excedence sets of suitable
pattern-avoiding subsets of the symmetric and hyperoctahedral groups.Comment: 15 pages, 2 figure
Decomposition and enumeration in partially ordered sets
Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999.Includes bibliographical references (p. 123-126).by Patricia Hersh.Ph.D
Refined enumeration of noncrossing chains and hook formulas
In the combinatorics of finite finite Coxeter groups, there is a simple
formula giving the number of maximal chains of noncrossing partitions. It is a
reinterpretation of a result by Deligne which is due to Chapoton, and the goal
of this article is to refine the formula. First, we prove a one-parameter
generalization, by the considering enumeration of noncrossing chains where we
put a weight on some relations. Second, we consider an equivalence relation on
noncrossing chains coming from the natural action of the group on set
partitions, and we show that each equivalence class has a simple generating
function. Using this we recover Postnikov's hook length formula in type A and
obtain a variant in type B.Comment: 18 pages. arXiv admin note: substantial text overlap with
arXiv:1304.090