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

TL;DR

This paper determines the extremal arrangements of cyclic continuants for Motzkin and Straus's sequences, providing unique maximizers and minimizers, and introduces an algorithm to construct special cyclic words with prescribed properties.

Contribution

It characterizes the extremal cyclic arrangements for regular and semi-regular continuants and develops an algorithm to construct all singular cyclic words with given Parikh vectors.

Findings

Unique maximizers and minimizers for cyclic continuants are identified.

A strong combinatorial condition called the singular property is established.

An algorithm for constructing all singular cyclic words with a given Parikh vector is developed.

Abstract

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.