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

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 $\Phi^{-1} (\sigma)$.