Dinv, area, and bounce for [formula omitted]-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 k→-Dyck paths by: i) giving a geometric construction for the bounce statistic of a 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 k→-Dyck path. Our bounce construction is inspired by Loehr's construction and Xin-Zhang's linear algorithm for inverting the sweep map on k→-Dyck paths. Our dinv interpretation is inspired by Garsia-Xin's visual proof of dinv-to-area result on rational Dyck paths.