Combinatorics of the zeta map on rational Dyck paths

An (a,b)(a,b)-Dyck path P   is a lattice path from (0,0)(0,0) to (b,a)(b,a) that stays above the line y=abx. The zeta map is a curious rule that maps the set of (a,b)(a,b)-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of P and zeta of P conjugate is enough to recover P.Our method begets an area-preserving involution χ   on the set of (a,b)(a,b)-Dyck paths when ζ   is a bijection, as well as a new method for calculating ζ−1ζ−1 on classical Dyck paths. For certain nice (a,b)(a,b)-Dyck paths we give an explicit formula for ζ−1ζ−1 and χ   and for additional (a,b)(a,b)-Dyck paths we discuss how to compute ζ−1ζ−1 and χ inductively.We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic δ   that can be used to recursively compute ζ−1ζ−1 and show that δ   is computable from ζ(P)ζ(P) in the Fuss–Catalan case.