Designing preference functions for de Bruijn sequences with forbidden words

A preference function provides a method to build periodic sequences by specifying a set of rules that determine which symbols are to be attempted before others, when the sequence is constructed one symbol at a time. The well-known prefer-one, prefer-opposite, and prefer-same binary de Bruijn sequences are all constructed using appropriate preference functions. In this article we provide some fairly general results that give conditions for a pair of an initial word and a preference function on a q-ary alphabet to produce sequences that include every pattern of given size n≥ 1 –except possibly some specified set of patterns. We provide several old and new constructions that showcase the flexibility of the results. Specifically, we give a construction for square-free and general separative de Bruijn sequences. The existence of these sequences was established more than a decade ago but nonconstructively. An important special case of these separative sequences produces universal cycles for permutations. We also build a preference function for binary de Bruijn sequences of patterns with a maximum density of ones. As for full de Bruijn sequences, the main result helps furnish a recursive construction from arbitrary cyclic permutations of q symbols. Finally, we build a preference function that extends a full de Bruijn sequence of order n into one of order n+ 1. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature

Stationary Distribution and Eigenvalues for a de Bruijn Process

We define a de Bruijn process with parameters n and L as a certain continuous-time Markov chain on the de Bruijn graph with words of length L over an n-letter alphabet as vertices. We determine explicitly its steady state distribution and its characteristic polynomial, which turns out to decompose into linear factors. In addition, we examine the stationary state of two specializations in detail. In the first one, the de Bruijn-Bernoulli process, this is a product measure. In the second one, the Skin-deep de Bruin process, the distribution has constant density but nontrivial correlation functions. The two point correlation function is determined using generating function techniques.Comment: Dedicated to Herb Wilf on the occasion of his 80th birthda

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