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 \(n\ge 1\)–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 \(n+1\).

首选项函数提供了一种构建周期性序列的方法，该方法通过指定一组规则来确定在一次构建一个符号时先尝试哪些符号。众所周知的prefer-one、prefer-opposite 和prefer-same 二元de Bruijn 序列都是使用适当的偏好函数构造的。在本文中，我们提供了一些相当一般的结果，这些结果给出了q -ary 字母表上的一对初始单词和偏好函数的条件，以生成包含给定大小\(n\ge 1\)的每个模式的序列——除了可能一些指定的模式集。我们提供了几种新旧结构，展示了结果的灵活性。具体来说，我们给出了一个无平方和一般的构造分离的 de Bruijn 序列。这些序列的存在是十多年前确定的，但非建设性的。这些分离序列的一个重要特例为置换产生了通用循环。我们还为具有最大密度的模式的二元 de Bruijn 序列构建了一个偏好函数。至于完整的 de Bruijn 序列，主要结果有助于从q符号的任意循环排列提供递归构造。最后，我们构建了一个偏好函数，将n阶的完整 de Bruijn 序列扩展为\(n+1\)阶之一。