On the extremal values of the cyclic continuants of Motzkin and Straus
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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<⋯ <a_k and a k-term partition P: n_1+n_2 + ⋯ + n_k=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 (n_1,n_2,…,n_k). We also show that the same is true for the minimizing arrangement for the semi-regular cyclic continuant K̇^↻(·). 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,…,a_k and not just on their relative order. We show that if a cyclic word x maximizes K̇^↻(·) 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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