A permutation of the elements of a finite set defines the partition n = x_1 + x_2 +...+ x_y of the number n of the elements of the set into the lengths of the y cycles of the permutation, and hence it defines the Young diagram (x_1 >= x_2 >= ... >= x_s). The talk describes the averaged statistics of the parameters of these diagrams for the n! permutations of the set and for the action of the so called "Arnold's cat map" (that I call Fibonacci operator), defined by the matrix 2 1 1 1 , on the finite torus Z_m x Z_m , consisting of n = m^2 points. The parameters of the Young diagram are: length x = x_1, width y, fullness lambda = n/(xy).