Right-jumps and pattern avoiding permutations

TL;DR

This paper investigates the process of right-jumps in permutations, characterizing the resulting pattern-avoiding classes, enumerating minimal forbidden patterns, and analyzing their asymptotic and probabilistic properties.

Contribution

It introduces a new permutation class generated by right-jumps, characterizes its basis, and derives its enumeration and asymptotic behavior, including a limit law for pattern structure.

Findings

The class of permutations after right-jumps forms a pattern-avoiding class.

The generating function for forbidden patterns is D-finite and satisfies a second-order differential equation.

Forbidden patterns typically have about (ln n)/√5 left-to-right maxima, with Gaussian fluctuations.

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 $(1+\sqrt 5)/2$) 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.