On the sweep map for fuss rational Dyck paths Abstract   Our main contribution here is the discovery of a new family of standard Young tableaux      T    n    k      which are in bijection with the family      D    m  ,  n      of Rational Dyck paths for    m    k  xc3x97  n  xc2xb1  1    (the so called xe2x80x9cFussxe2x80x9d 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 ThomasWilliam algorithm.

Subsequence containment by involutions Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in cal S_n which contain a given permutation tauincal S_k as a subsequence; this number depends on the patterns of the first j values of tau for 1leq jleq k. We then use this to define a partition of cal S_k, analogous to Wilfclasses in the study of pattern avoidance, and examine properties of this equivalence.  In the process, we show that a permutation tau_1ldotstau_k is layered iff, for 1leq jleq k, the pattern of tau_1ldotstau_j is an involution.  We also obtain a result of Sagan and Stanley counting the standard Young tableaux of size n which contain a fixed tableau of size k as a subtableau.