Ascent Sequences and Upper Triangular Matrices Containing Non-Negative Integers

Dukes, Mark; Parviainen, Robert

doi:10.37236/325

Cited by 47 publications

(82 citation statements)

References 5 publications

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“…Our proof here is the first direct approach based only on the decomposition of ascent sequences. The z = 1 case implies that max is a Stirling statistic on A n , proving a conjecture by Dukes and Parviainen [5,Conj. 13].…”

Section: Introductionmentioning

confidence: 53%

“…This paper is devoted to a systematic study of joint distributions of some classical word statistics, which we classify as Eulerian or Stirling statistics, on ascent sequences. In particular, two conjectures regarding ascent sequences due respectively to Dukes-Parviainen [5] and Levande [17] are solved. Our central contribution is the discovery of a new decomposition of ascent sequences.…”

Section: Introductionmentioning

confidence: 99%

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A new decomposition of ascent sequences and Euler–Stirling statistics

Jin

Zhao

et al. 2020

Journal of Combinatorial Theory, Series A

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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.

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Section: Introductionmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 99%

A new decomposition of ascent sequences and Euler–Stirling statistics

Jin

Zhao

et al. 2020

Journal of Combinatorial Theory, Series A

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“…Proof. Point (5) follows directly from point (3). Similarly, point (6) is an immediate consequence of the point (4) with q = 1.…”

Section: Example 21mentioning

confidence: 87%

“…For point (3), from the construction of the addition operation g, we see that the cell (x n + 1, dim(A)) is always a wNE cell. Moreover, there is a wNE-cell in row i of A if and only if there is a wNE-cell in row i of B and i < x n + 1.…”

Section: Lemma 33mentioning

confidence: 99%

Equidistributed Statistics on Fishburn Matrices and Permutations

Chen

Yan

Zhou

2019

Electron. J. Combin.

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“…For NODH interval orders, Fishburn called these characteristic matrices in [11]. They have been more recently studied by Dukes and Parviainen in [8]; Dukes et al in [6]; and Jelínek in [17]. In [17], Jelínek studied the class of what he calls Fishburn matrices that extend to the case where duplicated holdings are allowed.…”

Section: Background and Motivationmentioning

confidence: 99%

Hereditary Semiorders and Enumeration of Semiorders by Dimension

Keller

Young

2020

Electron. J. Combin.

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In 2010, Bousquet-Mélou et al. defined sequences of nonnegative integers called ascent sequences and showed that the ascent sequences of length n are in one-toone correspondence with the interval orders, i.e., the posets not containing the poset 2 + 2. Through the use of generating functions, this provided an answer to the longstanding open question of enumerating the (unlabeled) interval orders. A semiorder is an interval order having a representation in which all intervals have the same length. In terms of forbidden subposets, the semiorders exclude 2 + 2 and 1 + 3. The number of unlabeled semiorders on n points has long been known to be the n th Catalan number. However, describing the ascent sequences that correspond to the semiorders under the bijection of Bousquet-Mélou et al. has proved difficult. In this paper, we discuss a major part of the difficulty in this area: the ascent sequence corresponding to a semiorder may have an initial subsequence that corresponds to an interval order that is not a semiorder.We define the hereditary semiorders to be those corresponding to an ascent sequence for which every initial subsequence also corresponds to a semiorder. We provide a structural result that characterizes the hereditary semiorders and use this characterization to determine the ordinary generating function for hereditary semiorders. We also use our characterization of hereditary semiorders and the characterization of semiorders of dimension 3 given by Rabinovitch to provide a structural description of the semiorders of dimension at most 2. From this description, we are able to determine the ordinary generating for the semiorders of dimension at most 2.

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Ascent Sequences and Upper Triangular Matrices Containing Non-Negative Integers

Cited by 47 publications

References 5 publications

A new decomposition of ascent sequences and Euler–Stirling statistics

A new decomposition of ascent sequences and Euler–Stirling statistics

Equidistributed Statistics on Fishburn Matrices and Permutations

Hereditary Semiorders and Enumeration of Semiorders by Dimension

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