Enumerating $(\bf{2+2})$-free posets by indistinguishable elements

Dukes, Mark; Kitaev, Sergey; Remmel, Jeffrey B.; Steingrı́msson, Einar

doi:10.4310/joc.2011.v2.n1.a6

Cited by 25 publications

(38 citation statements)

References 11 publications

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“…We refine the set P n (12123) as follows. Given n ≥ 2 and 1 ≤ s < t ≤ n, let A n,t,s denote the subset of P n (12123) consisting of those partitions π = π 1 π 2 · · · π n having at least two distinct letters in which the left-most occurrence of the largest letter is at position t and the left-most occurrence of the second largest letter is at position s. For example, π = 123324425215 ∈ A 12,9,6 since the left-most occurrence of the largest letter, namely, 5, is at position 9 and the left-most occurrence of the second largest letter is at position 6. The array a n,t,s = |A n,t,s | is determined by the following recurrence.…”

Section: The Case 1012mentioning

confidence: 99%

Some enumerative results related to ascent sequences

Mansour

Shattuck

2014

Discrete Mathematics

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An ascent sequence is one consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it in the sequence. Ascent sequences have recently been shown to be related to (2+2)-free posets and a variety of other combinatorial structures. In this paper, we prove in the affirmative some recent conjectures concerning pattern avoidance for ascent sequences. Given a pattern τ , let Sτ (n) denote the set of ascent sequences of length n avoiding τ . Here, we show that the joint distribution of the statistic pair (asc, zeros) on S0012(n) is the same as (asc, RLmax) on the set of 132-avoiding permutations of length n. In particular, the ascent statistic on S0012(n) has the Narayana distribution. We also enumerate Sτ (n) when τ = 1012 and τ = 0123 and confirm the conjectured formulas in these cases. We combine combinatorial and algebraic techniques to prove our results, in two cases, making use of the kernel method. Finally, we discuss the case of avoiding 210 and determine two related recurrences.

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Section: The Case 1012mentioning

confidence: 99%

Some enumerative results related to ascent sequences

Mansour

Shattuck

2014

Discrete Mathematics

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“…Remarkably, the set of transitive web worlds is in one-to-one correspondence with the (2 + 2)-free posets. It seems likely that this connection will engender some interesting results, given the fast growing literature on these posets (see [1,2,4,6,5] for references).…”

Section: Introductionmentioning

confidence: 99%

Web worlds, web-colouring matrices, and web-mixing matrices

Dukes¹,

Gardi²,

Steingrı́msson³

et al. 2013

Journal of Combinatorial Theory, Series A

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We introduce a new combinatorial object called a web world that consists of a set of web diagrams. The diagrams of a web world are generalizations of graphs, and each is built on the same underlying graph. Instead of ordinary vertices the diagrams have pegs, and edges incident to a peg have different heights on the peg. The web world of a web diagram is the set of all web diagrams that result from permuting the order in which endpoints of edges appear on a peg. The motivation comes from particle physics, where web diagrams arise as particular types of Feynman diagrams describing scattering amplitudes in non-Abelian gauge (Yang-Mills) theories. To each web world we associate two matrices called the web-colouring matrix and web-mixing matrix. The entries of these matrices are indexed by ordered pairs of web diagrams (D1, D2), and are computed from those colourings of the edges of D1 that yield D2 under a transformation determined by each colouring.We show that colourings of a web diagram (whose constituent indecomposable diagrams are all unique) that lead to a reconstruction of the diagram are equivalent to order-preserving mappings of certain partially ordered sets (posets) that may be constructed from the web diagrams. For web worlds whose web graphs have all edge labels equal to 1, the diagonal entries of web-mixing and web-colouring matrices are obtained by summing certain polynomials determined by the descents in permutations in the Jordan-Hölder set of all linear extensions of the associated poset. We derive tri-variate generating generating functions for the number of web worlds according to three statistics and enumerate the number of different web diagrams in a web world. Three special web worlds are examined in great detail, and the traces of the web-mixing matrices calculated in each case.

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“…We have used Mathematica to compute the first few terms of A (p) (t, z) for p = 2, 3, 4: Next we can use the same techniques as in [4] to find the generating function for the number of primitive p-ascent sequences. That is, let r n,p denote the number of p-ascent sequences a of length n such that a has no consecutive repeated letters and a n,p denote the number of p-ascent sequences a of length n. If R (p) (t) = 1 + n≥1 r n,p t n and A (p) (t) = 1 + n≥1 a n,p t n , then it is easy to see that…”

Section: Resultsmentioning

confidence: 99%

A note on $p$-ascent sequences

Kitaev

Remmel

2017

Journal of Combinatorics

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A sequence (a 1 , . . . , a n ) of nonnegative integers is an ascent sequence if a 0 = 0 and for all i ≥ 2, a i is at most 1 plus the number of ascents in (a 1 , . . . , a i−1 ). Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes, and Kitaev in [1], who showed that these sequences of length n are in 1-to-1 correspondence with (2 + 2)-free posets of size n, which, in turn, are in 1-to-1 correspondence with interval orders of size n. Ascent sequences are also in bijection with several other classes of combinatorial objects including the set of upper triangular matrices with nonnegative integer entries such that no row or column contains all zeros, permutations that avoid a certain mesh pattern, and the set of Stoimenow matchings.In this paper, we introduce a generalization of ascent sequences, which we call p-ascent sequences, where p ≥ 1. A sequence (a 1 , . . . , a n ) of nonnegative integers is a p-ascent sequence if a 0 = 0 and for all i ≥ 2, a i is at most p plus the number of ascents in (a 1 , . . . , a i−1 ). Thus, in our terminology, ascent sequences are 1-ascent sequences. We generalize a result of the authors in [15] by enumerating p-ascent sequences with respect to the number of 0s. We also generalize a result of Dukes, Kitaev, Remmel, and Steingrímsson in [4] by finding the generating function for the number of p-ascent sequences which have no consecutive repeated elements. Finally, we initiate the study of pattern-avoiding p-ascent sequences.

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Enumerating $(\bf{2+2})$-free posets by indistinguishable elements

Cited by 25 publications

References 11 publications

Some enumerative results related to ascent sequences

Some enumerative results related to ascent sequences

Web worlds, web-colouring matrices, and web-mixing matrices

A note on $p$-ascent sequences

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