  We study the iteration of the process "a particle jumps 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 we enumerate these "forbidden minimal patterns" 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 nice differential equation of order~2. We give
some congruence properties for the coefficients of this generating function,
and we show that their asymptotics involves a rather unusual algebraic exponent
(the golden ratio $(1+\sqrt 5)/2$) and some unusual closed-form constants. We
end by proving a limit law: a forbidden pattern of length $n$ has typically
$(\ln n) /\sqrt{5}$ left-to-right maxima, with Gaussian fluctuations.

Discrete Mathematics;Combinatorics;Probability