Rearrangements of the symmetric group and enumerative properties of the tangent and secant numbers

Foata, Dominique; Strehl, Volker

doi:10.1007/bf01237393

Cited by 102 publications

(79 citation statements)

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“…These permutations were first studied by Foata, Strehl, and Schützenberger in a series of papers. In particular, Foata and Strehl defined in [5] and [6] a Z n 2 -action on S n where each orbit contains a unique André permutation of the first kind as a representative.…”

Section: Introductionmentioning

confidence: 99%

Permutation Trees and Variation Statistics

Hetyei

Reiner

1998

European Journal of Combinatorics

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In this paper we exploit binary tree representations of permutations to give a combinatorial proof of Purtill's result [8] thatwhere A n is the set of André permutations, v cd (σ ) is the cd-statistic of an André permutation and v ab (σ ) is the ab-statistic of a permutation. Using Purtill's proof as a motivation we introduce a new 'Foata-Strehl-like' action on permutations. This Z n−1 2 -action allows us to give an elementary proof of Purtill's theorem, and a bijection between André permutations of the first kind and alternating permutations starting with a descent. A modified version of our group action leads to a new class of André-like permutations with structure similar to that of simsun permutations.

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Section: Introductionmentioning

confidence: 99%

Permutation Trees and Variation Statistics

Hetyei

Reiner

1998

European Journal of Combinatorics

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“…An important fact is T K n+1 (1, −1) is equal to the number of 0-1-2 increasing trees in K n . This was first formulated by Foata [4] and was proved by Foata and Strehl [5]. Donaghey [3] gave a simple bijective proof for this fact.…”

Section: Introductionmentioning

confidence: 89%

Combinatorial interpretations for T_G(1, −1)

Yeh

2011

Journal of Graph Theory

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Let G be a connected and simple graph with vertex set {1, 2, . . . , n+1} and T G (x, y) the Tutte polynomial of G. In this paper, we give combinatorial interpretations for T G (1, −1). In particular, we give the definitions of even spanning tree and left spanning tree. We prove T G (1, −1) is the number of even-left spanning trees of G. We associate a permutation with a spanning forest of G and give the definition of odd G-permutations. We show T G (1, −1) is the number of odd G-permutations. We give a bijection from the set of odd K n+1 -permutations to the set of

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“…First, let x be a letter of a permutation σ = σ 1 σ 2 · · · σ n . The x-factorization of σ is defined to be the sequence (w 1 , w 2 , x, w 4 , w 5 ), where (1) the juxtaposition product w 1 w 2 xw 4 w 5 is equal to σ; (2) w 2 is the longest right factor of x 1 x 2 · · · x i−1 , all letters of which are greater than x; (3) w 4 is the longest left factor of x i+1 x i+2 · · · x n , all letters of which are greater than x. Foata and Strehl [13] introduced the involution ϕ x defined by…”

Section: )mentioning

confidence: 99%

“…(2) Consider the weight function W 2 . The weighted sum of the modified Foata-Strehl orbits Orb(σ) with the weight (1/2) pk(σ) is exactly the number of the original Foata-Strehl orbits, which is equal to the Euler numbers [13,14].…”

Section: )mentioning

confidence: 99%

Hankel continued fractions and Hankel determinants of the Euler numbers

Han

2020

Trans. Amer. Math. Soc.

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The Euler numbers occur in the Taylor expansion of tan(x) + sec(x). Since Stieltjes, continued fractions and Hankel determinants of the even Euler numbers, on the one hand, of the odd Euler numbers, on the other hand, have been widely studied separately. However, no continued fractions and Hankel determinants of the (mixed) Euler numbers have been obtained and explicitly calculated. The reason for that is that some Hankel determinants of the Euler numbers are null. This implies that the Jacobi continued fraction of the Euler numbers does not exist. In the present paper, this obstacle is bypassed by using the Hankel continued fraction, instead of the J-fraction. Consequently, an explicit formula for the Hankel determinants of the Euler numbers is being derived, as well as a full list of Hankel continued fractions and Hankel determinants involving Euler numbers. Finally, a new q-analog of the Euler numbers En(q) based on our continued fraction is proposed. We obtain an explicit formula for En(−1) and prove a conjecture by R. J. Mathar on these numbers.

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Rearrangements of the symmetric group and enumerative properties of the tangent and secant numbers

Cited by 102 publications

References 2 publications

Permutation Trees and Variation Statistics

Permutation Trees and Variation Statistics

Combinatorial interpretations for T_G(1, −1)

Hankel continued fractions and Hankel determinants of the Euler numbers

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Rearrangements of the symmetric group and enumerative properties of the tangent and secant numbers

Cited by 102 publications

References 2 publications

Permutation Trees and Variation Statistics

Permutation Trees and Variation Statistics

Combinatorial interpretations for TG(1, −1)

Hankel continued fractions and Hankel determinants of the Euler numbers

Contact Info

Product

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About

Combinatorial interpretations for T_G(1, −1)