On the extremal values of the cyclic continuants of Motzkin and Straus

In a 1983 paper, G. Ramharter asks what are the extremal arrangements for the cyclic analogues of the regular and semi-regular continuants first introduced by T.S. Motzkin and E.G. Straus in 1956. In this paper we answer this question by showing that for each set A consisting of positive integers 1 < a1 < a2 < · · · < ak and a k-term partition P : n1 + n2 + · · ·+ nk = n, there exists a unique (up to reversal) cyclic word x which maximizes (resp. minimizes) the regular cyclic continuant K (·) amongst all cyclic words over A with Parikh vector (n1, n2, . . . , nk). We also show that the same is true for the minimizing arrangement for the semi-regular cyclic continuant . K (·). As in the non-cyclic case, the main difficulty is to find the maximizing arrangement for . K (·), which is not unique in general and may depend on the integers a1, . . . , ak and not just on their relative order. We show that if a cyclic word x maximizes . K (·) amongst all permutations of x, then it verifies a strong combinatorial condition which we call the singular property. We develop an algorithm for constructing all singular cyclic words having a prescribed Parikh vector.