An efficient search algorithm for inverting the sweep map on rational Dyck paths

TL;DR

This paper introduces an efficient algorithm for inverting the sweep map on rational Dyck paths, providing a recursive solution for the Fuss case and a general search method for all cases.

Contribution

It presents a novel search algorithm to invert the sweep map on rational Dyck paths, extending invertibility results beyond the Fuss case.

Findings

Algorithm efficiently inverts the sweep map for all rational Dyck paths.

Recursive construction explicitly inverts the map in the Fuss case.

The method generalizes to non-Fuss cases using a $d$-array tree search.

Abstract

Given a coprime pair $(m,n)$ of positive integers, rational $(m,n)$-Dyck paths are lattice paths in the $m\times n$ rectangle that never go below the diagonal. The sweep map of a rational $(m,n)$-Dyck paths $D$ is the rational Dyck path $\Phi(D)$ obtained by sorting the steps of $D$ according to the ranks of their starting points, where the rank of $(a,b)$ is $bm-an$. It is conjectured to be a bijection, but to this date, $\Phi$ is only known to be bijective for the Fuss case ($m=kn\pm 1$). In this paper we give an efficient search algorithm for inverting the $\Phi$ map. Roughly speaking, given $\sigma\in \cal D_{m,n}$, by searching through a $d$-array tree of certain depth, we can output all $D$ such that $\Phi(D)=\sigma$, where $d$ is the remainder of $m$ when divided by $n$. In particular, we show that $\Phi$ is invertible for the Fuss case by giving a simple recursive construction for $\Phi^{-1} (\sigma)$.