论文标题
DINV，区域和弹跳，$ \ vec {k} $ - Dyck路径
Dinv, Area, and Bounce for $\vec{k}$-Dyck paths
论文作者
论文摘要
众所周知的$ Q，T $ -CATALAN序列具有两个组合解释，作为普通Dyck路径的加权总和：一个是Haglund的区域反弹配方，另一个是Haiman的Dinv-DiNv-Area公式。 Zeta图的构造是连接这两个公式的：这是从普通的Dyck路径到自身的两者，并且需要DINV到达区域，并弹跳。 Loehr以$ k $ -DYCK的路径扩展了这样的结果。 Zeta地图由Armstrong-Loehr-Warrington扩展了一类非常一般的路径。 在本文中，我们扩展了$ \ vec {k} $ - Dyck路径的DINV-AREA弹跳结果：i）给出了$ \ vec {k} $ dyck Path的弹跳统计量的几何结构，其中包括$ k $ -dyck $ -Dyck Paths和普通的dyck路径和普通的Dyck路径； ii）给出$ \ vec {k} $ - dyck路径的DINV统计量的几何解释。我们的弹跳结构的灵感来自Loehr的构造和Xin-Zhang的线性算法，用于在$ \ vec {k} $ -Dyck Paths上反转扫描图。我们的DINV解释灵感来自Garsia-Xin的DINV到区域的视觉证明，从而在理性的Dyck路径上产生了启发。
The well-known $q,t$-Catalan sequence has two combinatorial interpretations as weighted sums of ordinary Dyck paths: one is Haglund's area-bounce formula, and the other is Haiman's dinv-area formula. The zeta map was constructed to connect these two formulas: it is a bijection from ordinary Dyck paths to themselves, and it takes dinv to area, and area to bounce. Such a result was extended for $k$-Dyck paths by Loehr. The zeta map was extended by Armstrong-Loehr-Warrington for a very general class of paths. In this paper, We extend the dinv-area-bounce result for $\vec{k}$-Dyck paths by: i) giving a geometric construction for the bounce statistic of a $\vec{k}$-Dyck path, which includes the $k$-Dyck paths and ordinary Dyck paths as special cases; ii) giving a geometric interpretation of the dinv statistic of a $\vec{k}$-Dyck path. Our bounce construction is inspired by Loehr's construction and Xin-Zhang's linear algorithm for inverting the sweep map on $\vec{k}$-Dyck paths. Our dinv interpretation is inspired by Garsia-Xin's visual proof of dinv-to-area result on rational Dyck paths.
