On the Sweep Map for Fuss Rational Dyck Paths

Our main contribution here is the discovery of a new family of standard Young tableaux ${\cal T}^k_n$ which are in bijection with the family ${\cal D}_{m,n}$ of Rational Dyck paths for $m=k\times n\pm 1$ (the so called "Fuss" case). Using this family we give a new proof of the invertibility of the sweep map in the Fuss case by means of a very simple explicit algorithm. This new algorithm has running time $O(m+n)$. It is independent of the Thomas-William algorithm.