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 < . . . with values in the positive integers. Given a word w = w1 · · ·wn with wi ∈ N we define its multiplicity μ(w) as the number of times the value K(w) is assumed in the Abelian class X (w) of all permutations of the word w. We prove that there is an infinity of different lacunary alphabets of the form {b1 < · · · < bt < l + 1 < l + 2 < · · · < s} with bj , t, l, s ∈ N and s sufficiently large such that μ takes arbitrarily large values for words on these alphabets. The method of proof relies in part on a combinatorial characterisation of the word wmax in the class X (w) where K assumes its maximum. MSC: primary 11J70; secondary 68R15, 68W05, 05A20.