Factorization of Dyck words and the distribution of the divisors of an integer

TL;DR

This paper explores the connection between Dyck word factorizations and the divisor distribution of integers, providing a characterization of ensely divisible numbers through this combinatorial approach.

Contribution

It introduces a novel relationship between Dyck word factorizations and divisor distributions, offering a new characterization of ensely divisible numbers.

Findings

Factorization of Dyck words relates to divisor distribution of integers.

Characterization of ensely divisible numbers using Dyck word properties.

Establishment of a combinatorial framework for divisor analysis.

Abstract

In [CaballeroHooleyDelta], we associated a Dyck word $\langle\! \langle n \rangle\! \rangle_{\lambda}$ to any pair $(n, \lambda)$ consisting of an integer $n \geq 1$ and a real number $\lambda > 1$. The goal of the present paper is to show a relationship between the factorization of $\langle\! \langle n \rangle\! \rangle_{\lambda}$ as the concatenation of irreducible Dyck words and the distribution of the divisors of $n$. In particular, we will provide a characterization of $\lambda$-densely divisible numbers (these numbers were introduced in [castryck2014new]).