177 research outputs found
Nonhomogeneous parking functions and noncrossing partitions
For each skew shape we define a nonhomogeneous symmetric function,
generalizing a construction of Pak and Postnikov. In two special cases, we show
that the coefficients of this function when expanded in the complete
homogeneous basis are given in terms of the (reduced) type of -divisible
noncrossing partitions. Our work extends Haiman's notion of a parking function
symmetric function.Comment: 11 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
Enumeration of connected Catalan objects by type
Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted
plane trees are four classes of Catalan objects which carry a notion of type.
There exists a product formula which enumerates these objects according to
type. We define a notion of `connectivity' for these objects and prove an
analogous product formula which counts connected objects by type. Our proof of
this product formula is combinatorial and bijective. We extend this to a
product formula which counts objects with a fixed type and number of connected
components. We relate our product formulas to symmetric functions arising from
parking functions. We close by presenting an alternative proof of our product
formulas communicated to us by Christian Krattenthaler which uses generating
functions and Lagrange inversion
Parking Spaces
Let be a Weyl group with root lattice and Coxeter number . The
elements of the finite torus are called the -{\sf parking
functions}, and we call the permutation representation of on the set of
-parking functions the (standard) -{\sf parking space}. Parking spaces
have interesting connections to enumerative combinatorics, diagonal harmonics,
and rational Cherednik algebras. In this paper we define two new -parking
spaces, called the {\sf noncrossing parking space} and the {\sf algebraic
parking space}, with the following features: 1) They are defined more generally
for real reflection groups. 2) They carry not just -actions, but -actions, where is the cyclic subgroup of generated by a Coxeter
element. 3) In the crystallographic case, both are isomorphic to the standard
-parking space. Our Main Conjecture is that the two new parking spaces are
isomorphic to each other as permutation representations of . This
conjecture ties together several threads in the Catalan combinatorics of finite
reflection groups. We provide evidence for the conjecture, proofs of some
special cases, and suggest further directions for the theory.Comment: 49 pages, 10 figures, Version to appear in Advances in Mathematic
Cumulants and convolutions via Abel polynomials
We provide an unifying polynomial expression giving moments in terms of
cumulants, and viceversa, holding in the classical, boolean and free setting.
This is done by using a symbolic treatment of Abel polynomials. As a
by-product, we show that in the free cumulant theory the volume polynomial of
Pitman and Stanley plays the role of the complete Bell exponential polynomial
in the classical theory. Moreover via generalized Abel polynomials we construct
a new class of cumulants, including the classical, boolean and free ones, and
the convolutions linearized by them. Finally, via an umbral Fourier transform,
we state a explicit connection between boolean and free convolution
- …