Combinatorics of the zeta map on rational Dyck paths

TL;DR

This paper investigates the combinatorial properties of the zeta map on rational Dyck paths, providing progress towards its bijectivity, explicit formulas for inverse and involution, and new computational methods involving statistics and geometric interpretations.

Contribution

It introduces new methods for calculating the inverse and involution of the zeta map, and advances understanding of its bijectivity on rational Dyck paths.

Findings

Progress towards proving bijectivity of the zeta map.

Explicit formulas for inverse and involution in special cases.

New combinatorial methods using lasers and interval intersections.

Abstract

An $(a,b)$-Dyck path $P$ is a lattice path from $(0,0)$ to $(b,a)$ that stays above the line $y=\frac{a}{b}x$. The zeta map is a curious rule that maps the set of $(a,b)$-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 $\chi$ on the set of $(a,b)$-Dyck paths when $\zeta$ is a bijection, as well as a new method for calculating $\zeta^{-1}$ on classical Dyck paths. For certain nice $(a,b)$-Dyck paths we give an explicit formula for $\zeta^{-1}$ and $\chi$ and for additional $(a,b)$-Dyck paths we discuss how to compute $\zeta^{-1}$ and $\chi$ 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 $\delta$ that can be used to recursively compute $\zeta^{-1}$ and show that $\delta$ is computable from $\zeta(P)$ in the Fuss-Catalan case.