On the sweep map for fuss rational Dyck paths

Our main contribution here is the discovery of a new family of standard Young tableaux Tnk which are in bijection with the family Dm,n of Rational Dyck paths for m=k×n±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.