Partitions and partial matchings avoiding neighbor patterns

Chen, William Y. C.; Fan, Neil J. Y.; Zhao, Alina F. Y.

doi:10.1016/j.ejc.2011.09.039

Cited by 7 publications

(4 citation statements)

References 16 publications

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“…Another possibly fruitful path of generalisation could be to study the distribution of left and right nestings and crossings in partitions. Recent work of Chen et al [3] and of Yan and Xu [8] shows some progress in this direction.…”

Section: Acknowledgementmentioning

confidence: 99%

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Equidistributed statistics on matchings and permutations

Eriksen¹,

Sjöstrand²

2011

Preprint

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We show that the bistatistic of right nestings and right crossings in matchings without left nestings is equidistributed with the number of occurrences of two certain patterns in permutations, and furthermore that this equidistribution holds when refined to positions of these statistics in matchings and permutations. For this distribution we obtain a non-commutative generating function which specializes to Zagier's generating function for the Fishburn numbers after abelianization.As a special case we obtain proofs of two conjectures of Claesson and Linusson. Finally, we conjecture that our results can be generalized to involving left crossings of matchings too.

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Section: Acknowledgementmentioning

confidence: 99%

“…The map φ silly : T flat col-str → Ŝ is a bijection. Furthermore, we have(3) φ silly (T ⊕ T ′ ) = φ silly (T ) ⊕ φ silly (T ′ )…”

mentioning

confidence: 99%

Equidistributed statistics on matchings and permutations

Eriksen¹,

Sjöstrand²

2011

Preprint

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“…As a result, we derive the generating functions for partitions avoiding right crossings. Before we present the bijection, we should recall the notion of 2-paths defined by Chen et al [2]. Recall that a pair of two arcs (i, j) and (j, k) with i < j < k in the diagram of a partition is said to be a 2-path.…”

Section: Partitions With No Right Crossingsmentioning

confidence: 99%

“…Recently, Chen et al [2] derived the generating functions for partial matchings avoiding neighbor alignments and for partial matchings avoiding neighbor alignments and left nestings. They obtained the generating function for partitions avoiding right nestings by presenting a bijection between partial matchings avoiding three neighbor patterns ( left nestings, right nestings and neighbor alignments) and partitions avoiding right nestings.…”

Section: Introductionmentioning

confidence: 99%

On partitions avoiding right crossings

Yan,

2011

Preprint

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Recently, Chen et al. derived the generating function for partitions avoiding right nestings and posed the problem of finding the generating function for partitions avoiding right crossings. In this paper, we derive the generating function for partitions avoiding right crossings via an intermediate structure of partial matchings avoiding 2-right crossings and right nestings. We show that there is a bijection between partial matchings avoiding 2-right crossing and right nestings and partitions avoiding right crossings.

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Regular Simple Queues of Protein Contact Maps

2016

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A protein fold can be viewed as a self-avoiding walk in certain lattice model, and its contact map is a graph that represents the patterns of contacts in the fold. Goldman, Istrail, and Papadimitriou showed that a contact map in the 2D square lattice can be decomposed into at most two stacks and one queue. In the terminology of combinatorics, stacks and queues are noncrossing and nonnesting partitions, respectively. In this paper, we are concerned with 2-regular and 3-regular simple queues, for which the degree of each vertex is at most one and the arc lengths are at least 2 and 3, respectively. We show that 2-regular simple queues are in one-to-one correspondence with hill-free Motzkin paths, which have been enumerated by Barcucci, Pergola, Pinzani, and Rinaldi by using the Enumerating Combinatorial Objects method. We derive a recurrence relation for the generating function of Motzkin paths with [Formula: see text] peaks at level i, which reduces to the generating function for hill-free Motzkin paths. Moreover, we show that 3-regular simple queues are in one-to-one correspondence with Motzkin paths avoiding certain patterns. Then we obtain a formula for the generating function of 3-regular simple queues. Asymptotic formulas for 2-regular and 3-regular simple queues are derived based on the generating functions.

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Partitions and partial matchings avoiding neighbor patterns

Cited by 7 publications

References 16 publications

Equidistributed statistics on matchings and permutations

Equidistributed statistics on matchings and permutations

On partitions avoiding right crossings

Regular Simple Queues of Protein Contact Maps

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