Asymptotics of Permutations with Nearly Periodic Patterns of Rises and Falls

Bender, Edward A.; Helton, William; Richmond, L. Bruce

doi:10.37236/1733

Cited by 7 publications

(17 citation statements)

References 7 publications

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“…ε n of n rises and falls. The equivalence between both approaches has already been underlined in several works [8,21,23,24,25,27,28]. It will become clear in the following that each approach has its advantages: the probabilistic one is more suitable for analytical investigations, while the combinatorial one results in a simple recursive structure, lending itself to exact numerical calculations.…”

Section: Summary Of Resultsmentioning

confidence: 96%

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On the frequencies of patterns of rises and falls

Luck

2014

Physica A: Statistical Mechanics and its Applications

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We investigate the probability of observing a given pattern of n rises and falls in a random stationary data series. The data are modelled as a sequence of n + 1 independent and identically distributed random numbers. This probabilistic approach has a combinatorial equivalent, where the data are modelled by a random permutation on n + 1 objects. The probability of observing a long pattern of rises and falls decays exponentially with its length n in general. The associated decay rate α is interpreted as the embedding entropy of the pattern. This rate is evaluated exactly for all periodic patterns. In the most general case, it is expressed in terms of a determinant of generalized hyperbolic or trigonometric functions. Alternating patterns have the smallest rate α min = ln(π/2) = 0.451582 . . ., while other examples lead to arbitrarily large rates. The probabilities of observing uniformly chosen random patterns are demonstrated to obey multifractal statistics. The typical value α 0 = 0.806361 . . . of the rate plays the role of a Lyapunov exponent. A wide range of examples of patterns, either deterministic or random, is also investigated.

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Section: Summary Of Resultsmentioning

confidence: 96%

“…The differential equation (57) appears in [24] and [28]. Let us show that its solution can be simply expressed in terms of the generalized hyperbolic and trigonometric functions introduced in Appendix B.…”

Section: General Form Of the Solutionmentioning

confidence: 99%

On the frequencies of patterns of rises and falls

Luck

2014

Physica A: Statistical Mechanics and its Applications

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“…, it is possible to determine probabilistic quantities on S n by integrating certain functions that are constant on each region R σ . The appropriate functions for descent words were found in [10], yielding some new estimates like in [9] and [3]. The model of Ehrenborg, Levin and Readdy is presented in this paragraph, but in a modified way to focus only on the set of extreme cells E (as defined in Section 4.1).…”

Section: 1mentioning

confidence: 99%

“…The reading word σ(T ) of a standard tableau T is the permutation obtained by concatenating the rows of T from top to bottom. For example, the reading word of the standard tableau of Figure 8 is (8,11,2,5,9,1,3,4,6,7,10). By [31, Lemma A.1.1.10], P σ(T ) = T for all standard tableaux T .…”

Section: 2mentioning

confidence: 99%

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Zigzag diagrams and Martin boundary

Tarrago¹

2018

Ann. Probab.

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We investigate the asymptotic behavior of random paths on a graded graph which describes the subword order for words in two letters. This graph, denoted by Z, has been introduced by Viennot, who also discovered a remarkable bijection between paths on Z and sequences of permutations. Later on, Gnedin and Olshanski used this bijection to describe the set of Gibbs measures on this graph. Both authors also conjectured that the Martin boundary of Z should coincide with its minimal boundary. We give here a proof of this conjecture by describing the distribution of a large random path conditioned on having a prescribed endpoint. We also relate paths on the graph Z with paths on the Young lattice, and we finally give a central limit theorem for the Plancherel measure on the set of paths in Z. * Supported by the Deutsch-Französiche Hochschule MSC 2010 subject classifications: Primary 60C05, 60J45; secondary 05E99

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Exact and asymptotic enumeration of cyclic permutations according to descent set

Elizalde

Troyka

2019

Journal of Combinatorial Theory, Series A

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Using a result of Gessel and Reutenauer, we find a simple formula for the number of cyclic permutations with a given descent set, by expressing it in terms of ordinary descent numbers (i.e., those counting all permutations with a given descent set). We then use this formula to show that, for almost all sets I ⊆ [n − 1], the fraction of size-n permutations with descent set I which are n-cycles is asymptotically 1/n. As a special case, we recover a result of Stanley for alternating cycles. We also use our formula to count n-cycles with no two consecutive descents.

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Asymptotics of Permutations with Nearly Periodic Patterns of Rises and Falls

Cited by 7 publications

References 7 publications

On the frequencies of patterns of rises and falls

On the frequencies of patterns of rises and falls

Zigzag diagrams and Martin boundary

Exact and asymptotic enumeration of cyclic permutations according to descent set

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