From Permutations to Horizontal Visibility Patterns of Periodic Series

Periodic series of period T can be mapped into the set of permutations of [T−1]={1,2,3,…,T−1}. These permutations of period T can be classified according to the relative ordering of their elements by the horizontal visibility map. We prove that the number of horizontal visibility classes for each period T coincides with the number of triangulations of the polygon of T+1 vertices that, as is well known, is the Catalan number CT−1. We also study the robustness against Gaussian noise of the permutation patterns for each period and show that there are periodic permutations that better resist the increase of the variance of the noise.

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Snow Leopard Permutations and Their Even and Odd Threads

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.1279 ◽

2016 ◽

Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽

Author(s):

Eric S. Egge ◽

Kailee Rubin

Keyword(s):

Computer Science ◽

Pattern Avoidance ◽

Theoretical Computer Science ◽

Catalan Number ◽

The Other ◽

Discrete Mathematics ◽

Snow Leopard ◽

Theoretical Computer ◽

Set Of Permutations ◽

Motzkin Paths

Caffrey, Egge, Michel, Rubin and Ver Steegh recently introduced snow leopard permutations, which are the anti-Baxter permutations that are compatible with the doubly alternating Baxter permutations. Among other things, they showed that these permutations preserve parity, and that the number of snow leopard permutations of length $2n-1$ is the Catalan number $C_n$. In this paper we investigate the permutations that the snow leopard permutations induce on their even and odd entries; we call these the even threads and the odd threads, respectively. We give recursive bijections between these permutations and certain families of Catalan paths. We characterize the odd (resp. even) threads which form the other half of a snow leopard permutation whose even (resp. odd) thread is layered in terms of pattern avoidance, and we give a constructive bijection between the set of permutations of length $n$ which are both even threads and odd threads and the set of peakless Motzkin paths of length $n+1$. Comment: 25 pages, 6 figures. Version 3 is modified to use standard Discrete Mathematics and Theoretical Computer Science but is otherwise unchanged

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Alternating, Pattern-Avoiding Permutations

The Electronic Journal of Combinatorics ◽

10.37236/245 ◽

2009 ◽

Vol 16 (1) ◽

Cited By ~ 1

Author(s):

Joel Brewster Lewis

Keyword(s):

Permutation Patterns ◽

Alternating Permutations ◽

Set Of Permutations ◽

Pattern Avoiding Permutations

We study the problem of counting alternating permutations avoiding collections of permutation patterns including $132$. We construct a bijection between the set $S_n(132)$ of $132$-avoiding permutations and the set $A_{2n + 1}(132)$ of alternating, $132$-avoiding permutations. For every set $p_1, \ldots, p_k$ of patterns and certain related patterns $q_1, \ldots, q_k$, our bijection restricts to a bijection between $S_n(132, p_1, \ldots, p_k)$, the set of permutations avoiding $132$ and the $p_i$, and $A_{2n + 1}(132, q_1, \ldots, q_k)$, the set of alternating permutations avoiding $132$ and the $q_i$. This reduces the enumeration of the latter set to that of the former.

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Right-jumps and pattern avoiding permutations

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.1344 ◽

2017 ◽

Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽

Author(s):

Cyril Banderier ◽

Jean-Luc Baril ◽

Céline Moreira Dos Santos

Keyword(s):

Generating Function ◽

Analysis Of Algorithms ◽

Golden Ratio ◽

Permutation Patterns ◽

Exponential Generating Function ◽

Bivariate Exponential ◽

Set Of Permutations ◽

The Right ◽

Pattern Avoiding Permutations ◽

Congruence Properties

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. Comment: Corresponds to a work presented at the conferences Analysis of Algorithms (AofA'15) and Permutation Patterns'15. This arXiv revision just contains some cosmetic changes to fit to the journal style

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