Fishburn数及其精化Generating Functions的两种新解释
Two New Interpretations of the Fishburn Numbers and their Refined Generating Functions
摘要：
我们表明，两类组合对象-倒排表没有连续递减的数字和匹配没有2嵌套的子序列-列举的菲什伯恩数。特别是，我们给出了一个简单的双射匹配没有2套和反演表没有连续递减的子序列。然后，我们证明了使用对合原理，反演表没有减少连续数的子序列具有相同的生成函数的菲什伯恩数。Fishburn数之前已经被Bousquet-Mélou、Claesson、Dukes和Kitaev证明可以枚举$\textbf{(2+2)}$ -避免偏序集、没有左嵌套或右嵌套的匹配、避免特定模式的置换和所谓的上升序列，Dukes和Parviainen证明可以枚举具有非负元素和没有空行或列的上三角形矩阵。Claesson和Linusson证实他们也列举了没有2嵌套的匹配。使用这些新的解释的Fishburn数和另一个版本的对合，我们证明了两个细化Remmel和Kitaev的Fishburn生成函数的约束等式（也证明了使用矩阵的Jel 'ınek和严）。在附录中，我们陈述并证明了Claesson和Linusson的另一个猜想，给出了所有匹配集上左嵌套的分布。
英文摘要：
We show that two classes of combinatorial objects--inversion tables with no subsequence of decreasing consecutive numbers and matchings with no 2-nestings--are enumerated by the Fishburn numbers. In particular, we give a simple bijection between matchings with no 2-nestings and inversion tables with no subsequence of decreasing consecutive numbers. We then prove using the involution principle that inversion tables with no subsequence of decreasing consecutive numbers have the same generating function as the Fishburn numbers. The Fishburn numbers have previously been shown by Bousquet-Mélou, Claesson, Dukes and Kitaev to enumerate $\textbf{(2+2)}$-avoiding posets, matchings with no left- or right-nestings, permutations avoiding a particular pattern, and so-called ascent sequences, and by Dukes and Parviainen to enumerate upper triangular matrices with non-negative entries and no empty rows or columns. Claesson and Linusson conjectured they also enumerated matchings with no 2-nestings. Using these new interpretations of the Fishburn numbers and another version of the involution, we prove the conjectured equality (also proven using matrices by Jel\'ınek and by Yan) of two refinements by Remmel and Kitaev of the Fishburn generating function. In an appendix, we state and prove another conjecture of Claesson and Linusson giving the distribution of left-nestings over the set of all matchings.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
