More bijections for Entringer and Arnold families

Shin, Heon‐Cheol; Zeng, Jiang

doi:10.3934/era.2020111

Cited by 6 publications

(4 citation statements)

References 15 publications

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“…are introduced by Hoffman [9]. Following [14], a sequence of combinatorial objects (X n,k ) is called an Arnold family if |X n,k | = v n,k for 1 |k| n. Furthermore, it is a refined Arnold family if we can find a statistic over (X n,k ) such that V n,k is the enumerator with respect to this statistic. For example, for signed permutations, it is proved by Josuat-Vergés [10] that V n,k (t) for k > 0 (k < 0, respectively) counts the type B n snakes (D n , respectively) with respect to the statistic 'change of sign'.…”

Section: Arnold Family and Polynomial Refinementsmentioning

confidence: 99%

Three New Refined Arnold Families

Eu,

Kao

2023

Electron. J. Combin.

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The Springer numbers, introduced by Arnold, are generalizations of Euler numbers in the sense of Coxeter groups. They appear as the row sums of a double triangular array $(v_{n,k})$ of integers, $1\leq|k|\leq n$, defined recursively by a boustrophedon algorithm. We say a sequence of combinatorial objects $(X_{n,k})$ is an Arnold family if $X_{n,k}$ is counted by $v_{n,k}$. A polynomial refinement $V_{n,k}(t)$ of $v_{n,k}$, together with the combinatorial interpretations in several combinatorial structures was introduced by Eu and Fu recently. In this paper, we provide three new Arnold families of combinatorial objects, namely the cycle-up-down permutations, the valley signed permutations and Knuth's flip equivalences on permutations. We shall find corresponding statistics to realize the refined polynomial arrays.

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Section: Arnold Family and Polynomial Refinementsmentioning

confidence: 99%

Three New Refined Arnold Families

Eu,

Kao

2023

Electron. J. Combin.

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“…A bijection ζ 1 : ADI n+1 → RSI n . We describe a bijection between ADI n+1 and RSI n , which is extended from Shin and Zeng's [10] bijection between André permutation and Simsum permutations in S n .…”

Section: 3mentioning

confidence: 99%

“…Following [10], a sequence of sets (X n,k ) is called an Arnold family if #X n,k = v n,k for 1 ≤ |k| ≤ n. Josuat-Vergès, Novelli, and Thibon [9] studied β n -snakes from the point of view of combinatorial Hopf algebras, and came up with an Arnold family in terms of valley-signed permutations. Recently, Shin and Zeng [10] gave some Arnold families of objects: signed increasing 1-2 trees and a signed analogue of André permutations of the second kind.…”

Section: Introductionmentioning

confidence: 99%

Springer Numbers and Arnold Families Revisited

Eu¹,

Fu²

2021

Preprint

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For the calculation of Springer numbers (of root systems) of type Bn and Dn, Arnold introduced a signed analogue of alternating permutations, called βn-snakes, and derived recurrence relations for enumerating the βn-snakes starting with k. The results are presented in the form of double triangular arrays (v n,k ) of integers, 1 ≤ |k| ≤ n. An Arnold family is a sequence of sets of such objects as βn-snakes that are counted by (v n,k ). As a refinement of Arnold's result, we give analogous arrays of polynomials, defined by recurrence, for the calculation of the polynomials associated with successive derivatives of tan x and sec x, established by Hoffman. Moreover, we provide some new Arnold families of combinatorial objects that realize the polynomial arrays, which are signed variants of André permutations and Simsun permutations.

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“…In 1966, Entringer [2] enumerated the downup permutations according to the first element called Entringer numbers. Additionally, in the past two decades, a great deal of mathematical effort has been devoted to the study of Entringer numbers [7,11].…”

Section: Standard Puzzles Relating To Entringer Numbersmentioning

confidence: 99%

A skeleton model to enumerate standard puzzle sequences

Jia-Xi¹,

Ding²

2021

Preprint

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Guo-Niu Han [arXiv:2006.14070 [math.CO]] has introduced a new combinatorial object named standard puzzle. We use digraphs to show the relations between numbers in standard puzzles and propose a skeleton model. By this model, we solve the enumeration problem of over fifty thousand standard puzzle sequences. Most of them can be represented by classical numbers, such as Catalan numbers, double factorials, Secant numbers and so on. Also, we prove several identities in standard puzzle sequences.

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More bijections for Entringer and Arnold families

Cited by 6 publications

References 15 publications

Three New Refined Arnold Families

Three New Refined Arnold Families

Springer Numbers and Arnold Families Revisited

A skeleton model to enumerate standard puzzle sequences

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