Asymptotics and statistics on Fishburn matrices and their generalizations

Hwang, Hsien‐Kuei; Jin, Emma Yu

doi:10.48550/arxiv.1911.06690

Cited by 3 publications

(23 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In contrast to the Stirling statistics worked out in [14], which have typically logarithmic behaviors (logarithmic mean and logarithmic variance), the Eulerian statistics studied in this paper, namely, dimension distribution with fixed size, have asymptotically linear mean and linear variance (the corresponding dual statistic, size distribution of fixed dimension, is quadratic). Such a contrast is well known for statistics on permutations, but has remained mostly elusive on Fishburn structures.…”

mentioning

confidence: 89%

“…Similar to the classical integer partitions where 1 has a predominant frequency, the smallest nonzero entry in FMs also appears almost everywhere. But different from the exponential limit law of the occurrences of the smallest part in random integer partitions, the smallest entry in FMs has its occurrences following mostly (but not always) a Poisson limit law; see [14]. This indicates an even higher concentration of the smallest entry near its expected value in the context of random FMs.…”

mentioning

confidence: 93%

“…While there is a large literature on the combinatorial aspects of statistics over Fishburn structures, very few studies have been conducted on asymptotic and stochastic properties concerning structures of large size; see [6,23] and our previous paper [14]. Questions such as (see [15]) "what is the expected dimension of a random FM of size n when each of the size-n FMs is chosen with the same probability?"…”

mentioning

confidence: 99%

“…Since an FM of a given size can be viewed as an integer partition (but allowing 0 as entries) arranged on an upper-triangular matrix, there is yet a third class of Poisson statistics examined in detail in [14]: the number of occurrences of the smallest nonzero entry in the matrix. Similar to the classical integer partitions where 1 has a predominant frequency, the smallest nonzero entry in FMs also appears almost everywhere.…”

mentioning

confidence: 99%

“…The approach developed in [14] relies on a direct two-stage saddle-point method that is applied to the generating functions with a sum-of-product form, and is very powerful in that it is not only applicable to the asymptotics of a wide class of concrete examples, but also provides an effective means of understanding the limit laws of Stirling statistics. In the present paper, we further extend the same saddle-point approach to Eulerian statistics.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow

Hwang¹,

Jin²,

Schlosser³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

A. We establish the asymptotic normality of the dimension of large-size random Fishburn matrices by a complex-analytic approach. The corresponding dual problem of size distribution under large dimension is also addressed and follows a quadratic type normal limit law. These results represent the first of their kind and solve two open questions raised in the combinatorial literature. They are presented in a general framework where the entries of the Fishburn matrices are not limited to binary or nonnegative integers. The analytic saddle-point approach we apply, based on a powerful transformation for q-series due to Andrews and Jelínek, is also useful in solving a conjecture of Stoimenow in Vassiliev invariants. C 1. Introduction and main results 1.1. Dimension distribution of fixed-size FMs 1.2. Size distribution of fixed-dimension FMs 1.3. Asymptotic density of connected regular LCDs 2. The mean and the variance of the dimension 2.1. The generating functions of moments 2.2. Combinatorial interpretations 2.3. The two-stage saddle-point approach 2.4. Asymptotics of the moments 3. Dimension of random FMs 3.1. A better bivariate generating function 3.2. The exponential prototype 3.3. The factorial growth order 3.4. The exponential growth order 3.5. The asymptotic equivalent and the proof of Proposition 9 3.6. Proof of Theorem 1 4. Proof of the Stoimenow conjecture (Theorem 4) 5. Size distribution of random FMs 5.1. Asymptotic normality of the size 5.2. Andresen and Kjeldsen's 1976 paper 5.3. Some related OEIS sequences 6. Self-dual FMs and FMs without 1's 6.1. Dimension of self-dual FMs 6.2. FMs without 1's

show abstract

mentioning

confidence: 89%

mentioning

confidence: 93%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow

Hwang¹,

Jin²,

Schlosser³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Enumeration of Irredundant Forests

Ingels,

Azaïs

2020

Preprint

View full text Add to dashboard Cite

Reverse search is a convenient method for enumerating structured objects, that can be used both to address theoretical issues and to solve data mining problems. This method has already been successfully developed to handle unordered trees. If the literature proposes solutions to enumerate singletons of trees, we study in this article a more general, higher combinatorial problem, the enumeration of sets of trees -forests. By compressing each forest into a Directed Acyclic Graph (DAG), we develop a reverse search like method to enumerate DAG compressing forests. Remarkably, we prove that these DAG are in bijection with the row-Fishburn matrices, a well-studied class of combinatorial objects. In a second step, we derive our forest enumeration to provide algorithms for tackling two related problems : (i) the enumeration of "subforests" of a forest, and (ii) the frequent "subforest" mining problem. All the methods presented in this article enumerate each item uniquely, up to isomorphism.

show abstract

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

Asymptotics and statistics on Fishburn matrices and their generalizations

Abstract: 8.2. Self-dual Λ-Fishburn matrices with λ 1 > 0 45 8.3. Asymptotics of Λ-Fishburn matrices whose smallest nonzero entry is 2 47 8.4. Statistics on Λ-Fishburn matrices whose smallest nonzero entry is 2 49 9. Conclusions 51 Acknowledgements 52 References 52

Cited by 3 publications

References 43 publications

Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow

Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow

Enumeration of Irredundant Forests

Proof of a bi-symmetric septuple equidistribution on ascent sequences

Contact Info

Product

Resources

About