Crossings and Nestings in Tangled Diagrams

Chen, William Y. C.; Qin, Jing; Reidys, Christian M.

doi:10.37236/810

Cited by 23 publications

(49 citation statements)

References 13 publications

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“…enhanced) 3-crossings has been registered as A108304 (resp. A108307) in OEIS: 2,5,15,52,202,859,3930, . .…”

Section: Introductionmentioning

confidence: 99%

Restricted inversion sequences and enhanced 3-noncrossing partitions

Zhao

2018

European Journal of Combinatorics

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We prove a conjecture due independently to Yan and Martinez-Savage that asserts inversion sequences with no weakly decreasing subsequence of length 3 and enhanced 3-noncrossing partitions have the same cardinality. Our approach applies both the generating tree technique and the so-called obstinate kernel method developed by Bousquet-Mélou. One application of this equinumerosity is a discovery of an intriguing identity involving numbers of classical and enhanced 3-noncrossing partitions.

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“…enhanced) 3-crossings has been registered as A108304 (resp. A108307) in OEIS: 2,5,15,52,202,859,3930, . .…”

Section: Introductionmentioning

confidence: 99%

Restricted inversion sequences and enhanced 3-noncrossing partitions

Zhao

2018

European Journal of Combinatorics

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“…In Section 5, we briefly survey several colored variations of these results. Our main finding is that many colored objects−such as matchings [7], permutations [6,37], and tangled diagrams [8]−can be realized as colored set partitions Λ satisfying certain conditions on min(Λ) and max(Λ); see Propositions 5.6, 5.12, and 5.16. The techniques we used to prove Theorem 1.4 can therefore also be used to easily derive the symmetric joint distribution of crossing and nesting numbers in these cases.…”

Section: Resultsmentioning

confidence: 97%

“…In this final section we discuss a few variations of set partitions with natural notions of crossings, nestings, and colorings. Our discussion here is partially expository, surveying results from [6,7,8,9,37]. In each case our noncrossing and nonnesting colored objects are in bijection with certain classes of noncrossing and nonnesting colored set partitions.…”

Section: Some Extensionsmentioning

confidence: 98%

Crossings and Nestings in Colored Set Partitions

Marberg¹

2013

Electron. J. Combin.

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Chen, Deng, Du, Stanley, and Yan introduced the notion of k-crossings and k-nestings for set partitions, and proved that the sizes of the largest k-crossings and k-nestings in the partitions of an n-set possess a symmetric joint distribution. This work considers a generalization of these results to set partitions whose arcs are labeled by an r-element set (which we call r-colored set partitions). In this context, a k-crossing or k-nesting is a sequence of arcs, all with the same color, which form a k-crossing or k-nesting in the usual sense. After showing that the sizes of the largest crossings and nestings in colored set partitions likewise have a symmetric joint distribution, we consider several related enumeration problems. We prove that r-colored set partitions with no crossing arcs of the same color are in bijection with certain paths in N r , generalizing the correspondence between noncrossing (uncolored) set partitions and 2-Motzkin paths. Combining this with recent work of Bousquet-Mélou and Mishna affords a proof that the sequence counting noncrossing 2-colored set partitions is P-recursive. We also discuss how our methods extend to several variations of colored set partitions with analogous notions of crossings and nestings.

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“…These structures play a role in the enumeration of RNA molecules (see [4, Section 1.1] and references therein.) Instead of giving an in-depth presentation of tangled diagrams we refer to the papers [4,5] for details, and quote the following crucial observation by Chen et al [5, Observation 2, page 3]:…”

Section: K -Noncrossing Tangled Diagrams With Isolated Pointsmentioning

confidence: 99%

“…Special instances of our asymptotic formula for the total number of vicious walks in the lock step model have been established by Krattenthaler et al [16,17] and Rubey [21]. The growth order for the number of vicious walks in the lock step model with a free and point, and for the number of k-non-crossing tangled diagrams has been determined by Grabiner [11] and Chen et al [4], respectively. To the author's best knowledge, the asymptotics for the number of vicious walks in the random turns model seem to be new.…”

Section: Introductionmentioning

confidence: 95%

Asymptotics for the number of walks in a Weyl chamber of type B

Feierl

2012

Random Struct Algorithms

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ABSTRACT:We consider lattice walks in R k confined to the region 0 < x 1 < x 2 . . . < x k with fixed (but arbitrary) starting and end points. These walks are assumed to be such that their number can be counted using a reflection principle argument. The main results are asymptotic formulas for the total number of walks of length n with either a fixed or a free end point for a general class of walks as n tends to infinity. As applications, we find the asymptotics for the number of k-non-crossing tangled diagrams as well as asymptotics for two k-vicious walkers models subject to a wall restriction.

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Crossings and Nestings in Tangled Diagrams

Cited by 23 publications

References 13 publications

Restricted inversion sequences and enhanced 3-noncrossing partitions

Restricted inversion sequences and enhanced 3-noncrossing partitions

Crossings and Nestings in Colored Set Partitions

Asymptotics for the number of walks in a Weyl chamber of type B

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