Right-jumps and pattern avoiding permutations

Cyril Banderier (1) , Jean-Luc Baril (2) , Céline Moreira dos Santos (2)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 Le2i - Laboratoire Electronique, Informatique et Image [UMR6306]

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 + √ 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)/ √ 5 left-to-right maxima, with Gaussian fluctuations.

Mots clés: D-finite function; supercongruence; left-to-right maximum; insertion sort; generating function; analytic combinatorics; Permutation pattern

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]; Variables complexes [math.CV]; Probabilités [math.PR]; Théorie des nombres [math.NT]

Cyril Banderier, Jean-Luc Baril, Céline Moreira dos Santos. Right-jumps and pattern avoiding permutations. Discrete Mathematics and Theoretical Computer Science, 2017, 18 (2), pp.1-17. ⟨hal-01236582v4⟩