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Right-jumps and pattern avoiding permutations

Abstract : 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.
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Contributor : Cyril Banderier <>
Submitted on : Monday, March 11, 2019 - 6:41:48 AM
Last modification on : Monday, March 30, 2020 - 8:50:38 AM
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  • HAL Id : hal-01236582, version 4
  • ARXIV : 1512.02171


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



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