单词、停车函数和集合划分上组合统计量的洗牌兼容性
Shuffle-compatibility for combinatorial statistics on words, parking functions, and set partitions
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中文总结 AI 辅助
研究单词、停车函数和集合划分上统计量的洗牌兼容性，推广排列统计量的洗牌兼容性概念。系统回顾相关统计量，定义洗牌代数，其与多种组合霍普夫代数相关，能给出霍普夫代数基新解释及新基。
中文摘要 AI 辅助
我们引入了单词、停车函数和集合划分上统计量的（弱）洗牌兼容性概念，推广了Gessel和Zhuang关于排列统计量的洗牌兼容性。对于停车函数和集合划分，我们系统回顾了FindStat数据库（以及文献）中出现的统计量。我们进一步定义了由统计量诱导的等价类上（弱）洗牌兼容统计量的（移位）洗牌代数。这些代数与各种组合霍普夫代数密切相关，如QSym、FQSym、PQSym和NCSym。这些构造给出了各种霍普夫代数基的新组合解释，在某些情况下还给出了全新的基。
英文摘要
We introduce notions of (weak) shuffle-compatibility for statistics on words, parking functions, and set partitions, generalizing Gessel and Zhuang's shuffle-compatibility for statistics on permutations. For parking functions and set partitions, we perform a systematic review of statistics that appear in the FindStat database (as well as the literature). We further define (shifted) shuffle algebras of (weakly) shuffle-compatible statistics on the equivalence classes induced by the statistics. These algebras relate closely to various combinatorial Hopf algebras such as QSym, FQSym, PQSym, and NCSym. These constructions yield new combinatorial interpretations of various Hopf algebra bases and, in some cases, new bases entirely.
