The class of permutations that avoid the bivincular pattern \((231, \{1\},\{1\})\) is known to be enumerated by the Fishburn numbers. In this paper, we call them Fishburn permutations and study their pattern avoidance. For classical patterns of size 3, we give a complete enumerative picture for regular and indecomposable Fishburn permutations. For patterns of size 4, we focus on a Wilf equivalence class of Fishburn permutations that are enumerated by the Catalan numbers. In addition, we also discuss a class enumerated by the binomial transform of the Catalan numbers and give conjectures for other equivalence classes of pattern-avoiding Fishburn permutations.

中文翻译：

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关于避免模式的Fishburn排列

避免使用双目模式\（（（231，\ {1 \}，\ {1 \}）\））的排列类别已知为Fishburn数。在本文中，我们称它们为Fishburn排列并研究其模式避免。对于大小为3的经典模式，我们为规则的和不可分解的Fishburn排列给出了完整的枚举图。对于大小为4的模式，我们关注由加泰罗尼亚数字枚举的Fishburn排列的Wilf等价类。另外，我们还讨论了由加泰罗尼亚数的二项式变换枚举的一类，并为避免模式的菲什伯恩排列的其他等价类给出了猜想。