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

Abstract

In [CaballeroHooleyDelta], we associated a Dyck word $⟨⟨n⟩{⟩}_{\lambda }$ to any pair $(n,\lambda )$ consisting of an integer $n\ge 1$ and a real number $\lambda >1$. The goal of the present paper is to show a relationship between the factorization of $⟨⟨n⟩{⟩}_{\lambda }$ 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]).

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