Continuants with equal values, a combinatorial approach

TL;DR

This paper explores the combinatorial properties of regular continuants, focusing on the multiplicity of their values across permutations of words over certain alphabets, revealing infinitely many alphabets with unbounded multiplicities.

Contribution

It introduces a novel combinatorial approach to analyze the multiplicity of continuant values on words over lacunary alphabets, identifying conditions for arbitrarily large multiplicities.

Findings

Existence of infinitely many lacunary alphabets with unbounded multiplicities.

Characterization of words maximizing the continuant value within permutation classes.

Demonstration of arbitrarily large multiplicities for continuant values on these alphabets.

Abstract

A regular continuant is the denominator $K$ of a terminating regular continued fraction, interpreted as a function of the partial quotients. We regard $K$ as a function defined on the set of all finite words on the alphabet $1<2<3<\dots$ with values in the positive integers. Given a word $w=w_1\cdots w_n$ with $w_i\in\mathbb{N}$ we define its multiplicity $\mu(w)$ as the number of times the value $K(w)$ is assumed in the Abelian class $\mathcal{X}(w)$ of all permutations of the word $w.$ We prove that there is an infinity of different lacunary alphabets of the form $\{b_1<\dots <b_t<l+1<l+2<\dots <s\}$ with $b_j, t, l, s\in\mathbb{N}$ and $s$ sufficiently large such that $\mu$ takes arbitrarily large values for words on these alphabets. The method of proof relies in part on a combinatorial characterisation of the word $w_{max}$ in the class $\mathcal{X}(w)$ where $K$ assumes its maximum.