Equidistributed Statistics on Fishburn Matrices and Permutations

Chen, Dandan; Yan, Sherry H. F.; Zhou, Robin D. P.

doi:10.37236/7926

Cited by 5 publications

(3 citation statements)

References 12 publications

(28 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…were studied in [CYZ19,Jel15] Consequently, it follows directly from Theorem 1.8 that Corollary 1.14. There is a bijection between F n and itself such that the following symmetric distribution holds:…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

Proof of a bi-symmetric septuple equidistribution on ascent sequences

Jin¹,

Schlosser²

2023

Combinatorial Theory

View full text Add to dashboard Cite

It is well known since the seminal work by Bousquet-Mélou, Claesson, Dukes and Kitaev (2010) that certain refinements of the ascent sequences with respect to several natural statistics are in bijection with corresponding refinements of (2 + 2)-free posets and permutations that avoid a bi-vincular pattern. Different multiply-refined enumerations of ascent sequences and other bijectively equivalent structures have subsequently been extensively studied by various authors.In this paper, our main contributions are• a bijective proof of a bi-symmetric septuple equidistribution of Euler-Stirling statistics on ascent sequences, involving the number of ascents (asc), the number of repeated entries (rep), the number of zeros (zero), the number of maximal entries (max), the number of right-to-left minima (rmin) and two auxiliary statistics;• a new transformation formula for non-terminating basic hypergeometric 4 ϕ 3 series expanded as an analytic function in base q around q = 1, which is utilized to prove two (bi)-symmetric quadruple equidistributions on ascent sequences.A by-product of our findings includes the affirmation of a conjecture about the bi-symmetric equidistribution between the quadruples of Euler-Stirling statistics (asc, rep, zero, max) and (rep, asc, max, zero) on ascent sequences, that was motivated by a double Eulerian equidistribution due to Foata (1977) and recently proposed by Fu, Lin, Yan, Zhou and the first author (2018).

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 95%

Proof of a bi-symmetric septuple equidistribution on ascent sequences

Jin¹,

Schlosser²

2023

Combinatorial Theory

View full text Add to dashboard Cite

show abstract

“…A proof of Conjecture 1 using Theorem 2 and the machinery of basic hypergeometric series was recently found and will be featured in a separate paper [12]. It follows from Corollary 3 and Theorem 4 that the pair (rmin, zero) is symmetric on A n , which has an alternative proof provided by Chen, Yan and Zhou [3].…”

Section: Recent Developmentsmentioning

confidence: 98%

A new decomposition of ascent sequences and Euler–Stirling statistics

Jin

Zhao

et al. 2020

Journal of Combinatorial Theory, Series A

Self Cite

View full text Add to dashboard Cite

As shown by Bousquet-Mélou-Claesson- Dukes-Kitaev (2010), ascent sequences can be used to encode (2 + 2)-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power seriesIn this paper, we present a novel way to recursively decompose ascent sequences, which leads to:• a calculation of the Euler-Stirling distribution on ascent sequences, including the numbers of ascents (asc), repeated entries (rep), zeros (zero) and maximal entries (max).In particular, this confirms and extends Dukes and Parviainen's conjecture on the equidistribution of zero and max. • a far-reaching generalization of the generating function formula for (asc, zero) due toJelínek. This is accomplished via a bijective proof of the quadruple equidistribution of (asc, rep, zero, max) and (rep, asc, rmin, zero), where rmin denotes the right-to-left minima statistic of ascent sequences. • an extension of a conjecture posed by Levande, which asserts that the pair (asc, zero) on ascent sequences has the same distribution as the pair (rep, max) on (2 − 1)-avoiding inversion sequences. This is achieved via a decomposition of (2 − 1)-avoiding inversion sequences parallel to that of ascent sequences. This work is motivated by a double Eulerian equidistribution of Foata (1977) and a tempting bi-symmetry conjecture, which asserts that the quadruples (asc, rep, zero, max) and (rep, asc, max, zero) are equidistributed on ascent sequences.

show abstract

“…Beginning with Duncan and Steingrímsson [17], several additional authors have studied patternavoiding ascent sequences [5,7,11,21,27,28,29,30,31,33], statistics on ascent sequences [11,21,24], and connections between these sets and other combinatorial objects [7,37]. The study of pattern-avoiding Fishburn permutations was initiated by Gil and Weiner [23], though they are also used as a bridge in work of Chen, Yan, and Zhou [10].…”

Section: Introductionmentioning

confidence: 99%

Pattern-Avoiding Fishburn Permutations and Ascent Sequences

Egge¹

2022

Preprint

View full text Add to dashboard Cite

A Fishburn permutation is a permutation which avoids the bivincular pattern (231, {1}, {1}), while an ascent sequence is a sequence of nonnegative integers in which each entry is less than or equal to one more than the number of ascents to its left. Fishburn permutations and ascent sequences are linked by a bijection g of Bousquet-Mélou, Claesson, Dukes, and Kitaev. We write F n (σ 1 , . . . , σ k ) to denote the set of Fishburn permutations of length n which avoid each of σ 1 , . . . , σ k and we write A n (α 1 , . . . , α k ) to denote the set of ascent sequences which avoid each of α 1 , . . . , α k . We settle a conjecture of Gil and Weiner by showing that g restricts to a bijection between F n (3412) and A n (201). Building on work of Gil and Weiner, we use elementary techniques to enumerate F n (123) with respect to inversion number and number of left-to-right maxima, obtaining expressions in terms of q-binomial coefficients, and to enumerate F n (123, σ) for all σ. We use generating tree techniques to study the generating functions for F n (321, 1423), F n (321, 3124), and F n (321, 2143) with respect to inversion number and number of left-to-right maxima. We use these results to show |F n (321, 1423)| = |F n (321, 3124)| = F n+2 − n − 1, where F n is a Fibonacci number, and |F n (321, 2143)| = 2 n−1 . We conclude with a variety of conjectures and open problems.

show abstract

Equidistributed Statistics on Fishburn Matrices and Permutations

Abstract: Recently, Jelínek conjectured that there exists a bijection between certain restricted permutations and Fishburn matrices such that the bijection verifies the equidistribution of several statistics. The main objective of this paper is to establish such a bijection.

Cited by 5 publications

References 12 publications

Proof of a bi-symmetric septuple equidistribution on ascent sequences

Proof of a bi-symmetric septuple equidistribution on ascent sequences

A new decomposition of ascent sequences and Euler–Stirling statistics

Pattern-Avoiding Fishburn Permutations and Ascent Sequences

Contact Info

Product

Resources

About