Sur une extension des nombres de Genocchi

Dumont, Dominique; Randrianarivony, Arthur

doi:10.1016/s0195-6698(95)90053-5

Cited by 12 publications

(11 citation statements)

References 7 publications

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“…The strip decomposition of the Motzkin path P = uuhduudddudduuhdd drawn in Figure 5 is S(P ) = {(5, 1), (6,3), (7,6), (9,8), (10,10), (14,12), (15,14)}. It is known that every Motzkin path P can be determined from the set S(P ) satisfying ( 11) by reversing the above procedure ( see [6,Fig.…”

Section: Crossings Over S N (321 3142)mentioning

confidence: 99%

“…Using recursion ( 13), Dumont showed the following theorem. To prove it, we can we can inquire into the paper of Dumont and Randrianarivony [10].…”

Section: Crossings Over S N (321 3142)mentioning

confidence: 99%

“…15 12 16 13 14, we have R(σ) = {(5, 1),(6,3),(7,6),(9,8),(10,10),(14,12),(15,14)}. Thus, the canonical reduced decomposition of σ is σ 1 σ 2 • • • σ 7 , with σ 1 = s 5 s 4 s 3 s 2 s 1 , σ 2 = s 6 s 5 s 4 s 3 , σ 3 = s 7 s 6 , σ 4 = s 9 s 8 , σ 5 = s 10 , σ 6 = s 14 s 13 s 12 , and σ 7 = s 15 s 14 .The following theorem is an extension of [6, Thm.…”

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confidence: 99%

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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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Section: Crossings Over S N (321 3142)mentioning

confidence: 99%

“…Using recursion ( 13), Dumont showed the following theorem. To prove it, we can we can inquire into the paper of Dumont and Randrianarivony [10].…”

Section: Crossings Over S N (321 3142)mentioning

confidence: 99%

mentioning

confidence: 99%

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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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“…• if σ = 6 1 7 2 3 8 4 10 5 11 9 15 12 16 13 14, we have R(σ) = {(5, 1), (6, 3), (7,6), (9,8), (10,10), (14,12), (15,14)}.…”

Section: Crossings Over S N (321 3 142)mentioning

confidence: 99%

“…The strip decomposition of the Motzkin path P = uuhduudddudduuhdd drawn in Figure 5 is S(P) = {(5, 1), (6, 3), (7,6), (9,8), (10,10), (14,12), (15,14)}. It is known that every Motzkin path P can be determined from the set S(P) satisfying (4.1) by inversing the above procedure ( see [6,Fig.…”

Section: Crossings Over S N (321 3 142)mentioning

confidence: 99%

Crossings and nestings over some Motzkin objects and $q$-Motzkin numbers

Andriantsoa¹,

Rakotomamonjy²

2020

Preprint

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In this paper, we investigate on enumeration of some Motzkin objects according to the numbers of crossings and nestings. More precisely, we compute and express in terms of continued fractions the distributions of the statistics numbers of crossings and nestings over the sets of 4321-and 3412-involutions and the set of (321, 3 142)-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 Chen et al. between (321, 3 142)-avoiding permutations and Motzkin paths. Furthermore, we manipulate the obtained continued fractions to get the recursion formulas for the polynomial distributions of crossings and nestings and we discover that results involve two new q-Motzkin numbers.

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Une Famille de Polynomes Qui Interpole Plusieurs Suites Classiques de Nombres

Randrianarivony

Zeng

1996

Advances in Applied Mathematics

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We give a common polynomial extension of the Euler numbers, Genocchi numbers, Eulerian polynomials, the recent median Euler numbers. We first study some general algebraic properties of these polynomials, which include the continued fraction expansion of its ordinary generating function and, by establishing the connection with a generating function of some staircases introduced by Dumont, we get several combinatorial interpretations of these polynomials and then several new combinatorial interpretations of the above classical numbers. Finally, we study also a similar extension of Springer numbers.

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Sur une extension des nombres de Genocchi

Cited by 12 publications

References 7 publications

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

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

Crossings and nestings over some Motzkin objects and $q$-Motzkin numbers

Une Famille de Polynomes Qui Interpole Plusieurs Suites Classiques de Nombres

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