Restricted permutations refined by number of crossings and nestings

Rakotomamonjy, Paul M.

doi:10.1016/j.disc.2020.111950

Cited by 4 publications

(11 citation statements)

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“…Viennot [5], A. Randrianarivony [11,12], S. Corteel [3], S. Burril et al [2] to S. Corteel et al [4]. Recently, the first author of this paper introduced the study of this statistic on permutations avoiding a single pattern of length three [10]. This one is devoted on the distribution of crossings on permutations avoiding a pair of patterns of length three.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Various statistic-Wilf-equivalence classes for subset of patterns of length three are known in [1,6,8,10,13]. In [10], Rakotomamonjy particularly provided the Wilf-equivalence classes modulo cr for single pattern of length three. He proved bijectively that the only non singleton class is {132, 213, 321}, i.e.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Since every 321-avoiding permutations are nonesting (see [10]), we have cr(α j ) = inv(α j ) − exc(α j ) = (j − 1)(n − j) for all j. Using the fact that F 1 n (321, 213; q) = F n−1 (321, 213; q), we obtain F n (321, 213; q) = F n−1 (321, 213; q) + n ∑ j=2 q (j−1)(n−j) .…”

Section: Additional Resultsmentioning

confidence: 99%

“…Example: we have 3142 (2,3) = 43152 and 3142 −(2,3) = 23514. Now, we prove here a fundamental lemma which is a particular case of Lemma 3.7 in [10]. 1) .…”

Section: Remark 22 For Any Permutation σ the Index I Is A Lower Trans...mentioning

confidence: 92%

“…and α n+1 (σ) = 0 where δ is the usual Kronecker symbol. Now, let us recall some needed notations introduced in [10]. Given a permutation σ and two integers a and b, we denote by σ (a,b) the obtained permutation from σ by the following way:…”

Section: Remark 22 For Any Permutation σ the Index I Is A Lower Trans...mentioning

confidence: 99%

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Crossings over permutations avoiding some pairs of patterns of length three

Rakotomamonjy,

Andriantsoa,

Randrianarivony

2019

Preprint

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In this paper, we compute the distributions of the statistic number of crossings over permutations avoiding one of the pairs {321, 231}, {123, 132} and {123, 213}. The obtained results are new combinatorial interpretations of two known triangles in terms of restricted permutations statistic. For some pairs of three length-patterns, we find relationships between the polynomial distributions of the crossings over permutations that avoid the pairs containing the pattern 231 on the first hand and the pattern 312 on the other hand.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Additional Resultsmentioning

confidence: 99%

“…Example: we have 3142 (2,3) = 43152 and 3142 −(2,3) = 23514. Now, we prove here a fundamental lemma which is a particular case of Lemma 3.7 in [10]. 1) .…”

Section: Remark 22 For Any Permutation σ the Index I Is A Lower Trans...mentioning

confidence: 92%

Section: Remark 22 For Any Permutation σ the Index I Is A Lower Trans...mentioning

confidence: 99%

See 3 more Smart Citations

Crossings over permutations avoiding some pairs of patterns of length three

Rakotomamonjy,

Andriantsoa,

Randrianarivony

2019

Preprint

Self Cite

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Crossings and Nestings Over Some Motzkin Objects and $q$-Motzkin Numbers

Andriantsoa¹,

Rakotomamonjy²

2021

Electron. J. Combin.

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We examine the enumeration of certain Motzkin objects according to the numbers of crossings and nestings. With respect to continued fractions, we compute and express the distributions of the statistics of the numbers of crossings and nestings over three sets, namely the set of $4321$-avoiding involutions, the set of $3412$-avoiding involutions, and the set of $(321,3\bar{1}42)$-avoiding permutations. To get our results, we exploit the bijection of Biane restricted to the sets of $4321$- and $3412$-avoiding involutions which was characterized by Barnabei et al. and the bijection between $(321,3\bar{1}42)$-avoiding permutations and Motzkin paths, presented by Chen et al.. Furthermore, we manipulate the obtained continued fractions to get the recursion formulas for the polynomial distributions of crossings and nestings, and it follows that the results involve two new $q$-Motzkin numbers.

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