Combinatorics of the zeta map on rational Dyck paths

An pa, bq-Dyck path P is a lattice path from p0, 0q to pb, aq that stays above the line y “ a b x. The zeta map is a curious rule that maps the set of pa, bq-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 pa, bq-Dyck paths when ζ is a bijection, as well as a new method for calculating ζ ́1 on classical Dyck paths. For certain nice pa, bq-Dyck paths we give an explicit formula for ζ ́1 and χ and for additional pa, bq-Dyck paths we discuss how to compute ζ ́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 ́1 and show that δ is computable from ζpP q in the Fuss-Catalan case.