Enumeration of connected Catalan objects by type

TL;DR

This paper introduces a combinatorial and bijective proof for a product formula counting connected Catalan objects by type, extending previous formulas and relating them to symmetric functions and parking functions.

Contribution

It defines a notion of connectivity for Catalan objects and proves a new product formula for counting connected objects by type, with extensions and alternative proofs.

Findings

Established a product formula for connected Catalan objects by type

Extended the formula to objects with fixed type and number of components

Connected the formulas to symmetric functions and parking functions

Abstract

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.