Right-jumps & pattern avoiding permutations

We study the iteration of the process of moving values to the right in permutations. We prove that the set of permutations obtained in this model after a given number of iterations from the identity is a class of pattern avoiding permutations. We characterize the elements of the basis of this class and enumerate it by giving their bivariate exponential generating function: we achieve this via a catalytic variable, the number of left-to-right maxima. We show that this generating function is a D-finite function satisfying a differential equation of order 2. We give some congruence properties for the coefficients of this generating function, and show that their asymptotics involves a rather unusual algebraic exponent (the golden ratio (1 + root 5)/2) and some closed-form constants. We end by proving a limit law: a forbidden pattern of length n has typically (ln n)/root 5 left-to-right maxima, with Gaussian fluctuations.