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 $$\sqrt{\frac{n}{\lambda }}<d\le \sqrt{\lambda n}.$$ We will prove that there are integers having an arbitrarily large number of $\lambda$-middle divisors. Consider the word $$⟨⟨n⟩{⟩}_{\lambda }:=w_1{w}_2...w_k\in \{a,b{\}}^{\ast },$$ given by $$w_i:=\{\begin{array}{cl}a& \text{if }{u}_i\in D_n\backslash \left(\lambda D_n\right),\\ b& \text{if }{u}_i\in \left(\lambda D_n\right)\backslash D_n,\end{array}$$ where $D_n$ is the set of divisors of $n$, $\lambda D_n:=\{\lambda d:d\in D_n\}$ and $u_1,u_2,...,u_k$ are the elements of the symmetric difference $D_n△\lambda D_n$ written in increasing order. We will prove that the language $${\mathcal{L}}_{\lambda }:=\left\{⟨⟨n⟩{⟩}_{\lambda }:n\in {\mathbb{Z}}_{\ge 1}\right\}$$ contains Dyck words having an arbitrarily large number of centered tunnels. We will show a connection between both results.

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