Counting closed walks in infinite regular trees using Catalan and Borel's triangles

TL;DR

This paper explores counting closed walks in infinite regular trees by employing Catalan's and Borel's triangles, revealing new combinatorial structures linked to these mathematical arrays.

Contribution

It introduces a novel combinatorial interpretation of Catalan's and Borel's triangles in the context of counting closed walks in regular trees.

Findings

Closed walks are counted using Catalan's triangle.

Borel's triangle also enumerates these walks.

New combinatorial structures are identified.

Abstract

We count the number of closed walks on a vertex in a regular tree using the Catalan's triangle and also the Borel's triangle, showing another combinatorial structure counted by these two array of numbers.