Dinv, Area, and Bounce for $\vec{k}$-Dyck paths

TL;DR

This paper extends the dinv-area-bounce relationship to $oldsymbol{k}$-Dyck paths by providing geometric constructions and interpretations for bounce and dinv statistics, generalizing previous results for ordinary and $k$-Dyck paths.

Contribution

It introduces geometric methods to define bounce and dinv statistics for $oldsymbol{k}$-Dyck paths, broadening the combinatorial understanding of these paths.

Findings

Geometric construction of bounce statistic for $oldsymbol{k}$-Dyck paths

Geometric interpretation of dinv statistic for $oldsymbol{k}$-Dyck paths

Unified framework encompassing ordinary, $k$-Dyck, and $oldsymbol{k}$-Dyck paths

Abstract

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.