Continuants with equal values, a combinatorial approach

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.