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 $1<a_1<a_2<\cdots <a_k$ and a $k$-term partition $P: n_1+n_2 + \cdots + n_k=n$, there exists a unique (up to reversal) cyclic word $x$ which maximizes (resp. minimizes) the regular cyclic continuant $K^{\circlearrowright}(\cdot)$ amongst all cyclic words over $A$ with Parikh vector $(n_1,n_2,\ldots,n_k)$. We also show that the same is true for the minimizing arrangement for the semi-regular cyclic continuant $\dot K^{\circlearrowright}(\cdot)$. As in the non-cyclic case, the main difficulty is to find the maximizing arrangement for the semi-regular continuant, which is not unique in general and may depend on the integers $a_1,\ldots,a_k$ and not just on their relative order. We show that if a cyclic word $x$ maximizes $\dot K^{\circlearrowright}(\cdot)$ 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.