论文标题
精制的晶格路径枚举和组合互惠
Refined Lattice Path Enumeration and Combinatorial Reciprocity
论文作者
论文摘要
众所周知，Fuß-Narayana数字计算了固定高度和固定山谷的$ M $ -DYCK路径。在本文中，我们考虑了至少$ t $ north步骤开头的$ M $ -DYCK路径。我们为具有固定高度，固定数量的回报次数和（i）固定数量的阀门的此类路径的数量提供了精确公式，（ii）固定数量的阀门数量为$ x $ - 坐标为$ m $ $ m $和（iii）固定数量的带有$ x $ coortion的valleys，不可除以$ m $。枚举（ii）合并意识到了$ h $三角形在最近的Krattenthaler文章中出现的$ H $三角形，并在某些抛物线非交叉分区的背景下（Algebr。Comb。5，2022）出现（代数Comb。5，2022）。通过由于肖普顿而引起的转换公式，我们为相关的$ f $ -triangle提供了明确的公式。我们通过广义的schröder路径以及某些超平面布置中的平地在跨驱动方面认识到这种多项式组合。在此过程中，我们展示了两个新的组合互惠结果。
It is well known that the set of $m$-Dyck paths with a fixed height and a fixed amount of valleys is counted by the Fuß-Narayana numbers. In this article, we consider the set of $m$-Dyck paths that start with at least $t$ north steps. We give exact formulas for the number of such paths with fixed height, fixed number of returns and (i) fixed number of valleys, (ii) fixed number of valleys with $x$-coordinate divisible by $m$ and (iii) fixed number of valleys with $x$-coordinate not divisible by $m$. The enumeration (ii) combinatorially realizes the $H$-triangle appearing in a recent article of Krattenthaler and the first author (Algebr. Comb. 5, 2022) in the context of certain parabolic noncrossing partitions. Through a transformation formula due to Chapoton, we give an explicit formula for the associated $F$-triangle. We realize this polynomial combinatorially by means of generalized Schröder paths as well as flats in certain hyperplane arrangements. Along the way we exhibit two new combinatorial reciprocity results.
