Middle divisors and $\sigma$-palindromic Dyck words

Abstract

Given a real number $\lambda > 1$, we say that $d|n$ is a $\lambda$-middle divisor of $n$ if $$ \sqrt{\frac{n}{\lambda}} < d \leq \sqrt{\lambda n}. $$ We will prove that there are integers having an arbitrarily large number of $\lambda$-middle divisors. Consider the word $$ \langle\! \langle n \rangle\! \rangle_{\lambda} := w_1 w_2 ... w_k \in \{a,b\}^{\ast}, $$ given by $$ w_i := \left\{ \begin{array}{c l} a & \textrm{if } u_i \in D_n \backslash \left(\lambda D_n\right), \\ b & \textrm{if } u_i \in \left(\lambda D_n\right)\backslash D_n, \end{array} \right. $$ where $D_n$ is the set of divisors of $n$, $\lambda D_n := \{\lambda d: \quad d \in D_n\}$ and $u_1, u_2, ..., u_k$ are the elements of the symmetric difference $D_n \triangle \lambda D_n$ written in increasing order. We will prove that the language $$ \mathcal{L}_{\lambda} := \left\{\langle\! \langle n \rangle\! \rangle_{\lambda} : \quad n \in \mathbb{Z}_{\geq 1} \right\} $$ contains Dyck words having an arbitrarily large number of centered tunnels. We will show a connection between both results.