Counting with Borel's Triangle

TL;DR

This paper explores combinatorial interpretations of Borel's triangle, relating it to lattice paths, trees, and permutations, and derives a functional equation to analyze these structures.

Contribution

It provides new combinatorial interpretations of Borel's triangle and introduces a functional equation for analyzing related combinatorial structures.

Findings

Borel's triangle relates to lattice paths, binary trees, and pattern-avoiding permutations.

A functional equation for Borel's triangle is derived.

Various combinatorial models for Borel's triangle are presented.

Abstract

Borel's triangle is an array of integers closely related to the classical Catalan numbers. In this paper we study combinatorial statistics counted by Borel's triangle. We present various combinatorial interpretations of Borel's triangle in terms of lattice paths, binary trees, and pattern avoiding permutations and matchings, and derive a functional equation that is useful in analyzing the involved structures.