Nonhomogeneous parking functions and noncrossing partitions

TL;DR

This paper introduces a new class of nonhomogeneous symmetric functions associated with skew shapes, linking them to noncrossing partitions and extending Haiman's parking function symmetric functions.

Contribution

It generalizes the construction of parking function symmetric functions and connects these to noncrossing partitions through explicit coefficient formulas.

Findings

Coefficients relate to the type of k-divisible noncrossing partitions

Extension of Haiman's parking function symmetric functions

Provides new combinatorial interpretations for symmetric functions

Abstract

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 $k$-divisible noncrossing partitions. Our work extends Haiman's notion of a parking function symmetric function.