Subsemigroup of Tn assigned to a fixed point free permutation

Assume that Tn is the full transformation semigroup of X = {1,..., n} and Sn. Tn is the symmetric group. For s. Sn and i. X we define fi, s. Tn by fi, s (i) = s(i), and fi, s ( j) = j for j = i. Let S be the subsemigroup of Tn generated by idempotents f1, s,..., fn, s . Let a. S. We study the sequence (card F(ak))k.N, where F(ak) is the set of fixed points of ak. We will focus on the case s = (1 2... n). Our motivation comes from a geometric method for detecting chaotic dynamics based on the Wa.zewski retract principle.