Partitions, pairs of trees and Catalan numbers

TL;DR

This paper provides a closed-form description and bijections for families of partitions related to Catalan and ballot numbers, extending previous recursive algorithms to explicit combinatorial structures.

Contribution

It introduces a closed description for partition families associated with Catalan and ballot numbers and establishes bijections with tree and forest structures.

Findings

Closed-form descriptions of partition families

Bijections between partitions and Catalan/ballot trees

Enhanced understanding of partition enumeration

Abstract

In "Square partitions and Catalan numbers" (arXiv0912.4983), Bennett et al. presented a recursive algorithm to create a family of partitions from one or several partitions. They were mainly interested in the cases when we begin with a single square partition or with several partitions with only one part. The cardinalities of those families of partitions are the Catalan and ballot numbers, respectively. In this paper we present a closed description for those families. We also present bijections between those sets of partitions and sets of trees and forests enumerated by the Catalan an ballot numbers.