Alternating, Pattern-Avoiding Permutations

Joel Brewster Lewis

doi:10.37236/245

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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Mesh Patterns and the Expansion of Permutation Statistics as Sums of Permutation Patterns

The Electronic Journal of Combinatorics ◽

10.37236/2001 ◽

2011 ◽

Vol 18 (2) ◽

Cited By ~ 19

Author(s):

Petter Brändén ◽

Anders Claesson

Keyword(s):

Fixed Points ◽

Relative Position ◽

Permutation Statistics ◽

Permutation Patterns ◽

Alternating Permutations ◽

Major Index ◽

Permutation Pattern ◽

New Type ◽

Permutation Statistic ◽

Infinite Linear

Any permutation statistic $f:{\mathfrak{S}}\to{\mathbb C}$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(\pi,R)$ is an occurrence of the permutation pattern $\pi$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(\pi,R)$, we have $\lambda_p(\tau) = (-1)^{|\tau|-|\pi|}{p}^{\star}(\tau)$ where ${p}^{\star}=(\pi,R^c)$ is the mesh pattern with the same underlying permutation as $p$ but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

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Beyond Alternating Permutations: Pattern Avoidance in Young Diagrams and Tableaux

The Electronic Journal of Combinatorics ◽

10.37236/3246 ◽

2013 ◽

Vol 20 (4) ◽

Author(s):

Nihal Gowravaram ◽

Ravi Jagadeesan

Keyword(s):

Pattern Avoidance ◽

General Context ◽

Young Tableaux ◽

Young Diagrams ◽

Alternating Permutations ◽

Equivalence Type ◽

Shape Equivalence ◽

Pattern Avoiding Permutations ◽

Wilf Equivalence

We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young tableaux, that is, permutations that have $k-1$ ascents followed by a descent, followed by $k-1$ ascents, et cetera. We determine the relative sizes of the sets of pattern-avoiding $(k-1)$-ascent permutations in terms of the forbidden pattern. Furthermore, inequalities in the sizes of sets of pattern-avoiding permutations in this context arise from further extensions of shape-equivalence type enumerations. This paper is the first of a two-paper series presenting the work of Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (arXiv:1301.6796v1). The second in the series is Ascent-descent Young diagrams and pattern avoidance in alternating permutations (by the second author, submitted).

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From Permutations to Horizontal Visibility Patterns of Periodic Series

Engineering Proceedings ◽

10.3390/engproc2021005009 ◽

2021 ◽

Vol 5 (1) ◽

pp. 9

Author(s):

Francisco J. Muñoz ◽

Juan Carlos Nuño

Keyword(s):

Gaussian Noise ◽

Catalan Number ◽

Permutation Patterns ◽

Horizontal Visibility ◽

Visibility Map ◽

Set Of Permutations

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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Large Deviations and Ratio Limit Theorems for Pattern-Avoiding Permutations

Combinatorics Probability Computing ◽

10.1017/s0963548313000576 ◽

2013 ◽

Vol 23 (2) ◽

pp. 161-200 ◽

Cited By ~ 5

Author(s):

MAHSHID ATAPOUR ◽

NEAL MADRAS

Keyword(s):

Large Deviations ◽

Limit Theorems ◽

Large Deviation ◽

Corner Points ◽

Set Of Permutations ◽

Ratio Limit Theorems ◽

Ratio Limit ◽

Left Corner ◽

To Come ◽

Pattern Avoiding Permutations

For a fixed permutation τ, let$\mathcal{S}_N(\tau)$be the set of permutations onNelements that avoid the pattern τ. Madras and Liu (2010) conjectured that$\lim_{N\rightarrow\infty}\frac{|\mathcal{S}_{N+1}(\tau)|}{ |\mathcal{S}_N(\tau)|}$exists; if it does, it must equal the Stanley–Wilf limit. We prove the conjecture for every permutation τ of length 5 or less, as well as for some longer cases (including 704 of the 720 permutations of length 6). We also consider permutations drawn at random from$\mathcal{S}_N(\tau)$, and we investigate properties of their graphs (viewing permutations as functions on {1,. . .,N}) scaled down to the unit square [0,1]2. We prove exact large deviation results for these graphs when τ has length 3; it follows, for example, that it is exponentially unlikely for a random 312-avoiding permutation to have points above the diagonal strip |y−x| < ε, but not unlikely to have points below the strip. For general τ, we show that some neighbourhood of the upper left corner of [0,1]2is exponentially unlikely to contain a point of the graph if and only if τ starts with its largest element. For patterns such as τ=4231 we establish that this neighbourhood can be extended along the sides of [0,1]2to come arbitrarily close to the corner points (0,0) and (1,1), as simulations had suggested.

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