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.
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