论文标题
图案HOPF代数的反模式公式
Antipode formulas for pattern Hopf algebras
论文作者
论文摘要
置换模式HOPF代数是一个交换性过滤和连接的Hopf代数。它的产品结构源于计数置换的模式，将系数解释为排列的准船。 HOPF代数被证明是一个自由的交换代数，并通过具有限制的物种来融入模式Hopf代数的一般框架。 在本文中，我们介绍了无取消和无分组的公式，用于置换模式的Hopf代数。为了获得此公式，我们使用了Benedetti和Sagan的流行签名转换方法。该公式在多项式不变式上应用于排列，特别是用于获得互惠定理。在途中，我们还介绍了包装的单词模式Hopf代数，并为其反座模式提供了一个公式。 这里讨论了其他模式代数，特别是关于停车功能，该功能恢复了Adeniran和Pudwell以及Qiu和Remmel最近研究的概念。
The permutation pattern Hopf algebra is a commutative filtered and connected Hopf algebra. Its product structure stems from counting patterns of a permutation, interpreting the coefficients as permutation quasi-shuffles. The Hopf algebra was shown to be a free commutative algebra and to fit into a general framework of pattern Hopf algebras, via species with restrictions. In this paper we introduce the cancellation-free and grouping-free formula for the antipode of the permutation pattern Hopf algebra. To obtain this formula, we use the popular sign-reversing involution method, by Benedetti and Sagan. This formula has applications on polynomial invariants on permutations, in particular for obtaining reciprocity theorems. On our way, we also introduce the packed word patterns Hopf algebra and present a formula for its antipode. Other pattern algebras are discussed here, notably on parking functions, which recovers notions recently studied by Adeniran and Pudwell, and by Qiu and Remmel.
