K-Noncrossing Trees and K-Proper Trees

Pang, Sabrina X. M.; Lv, Lun

doi:10.1109/iciecs.2010.5677757

Cited by 4 publications

(3 citation statements)

References 7 publications

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“…that make the optimization problems difficult to handle and can lead an optimizer to a suboptimal region in the search space. Simulated annealing, Monte-Carlo sampling, Stochastic tunneling, Parallel tempering, Stochastic Hill Climbing, PSO, GA, Evolution Strategies, Memetic Algorithms, Differential Evolution are some examples of stochastic algorithms [9,10,11,12,13,2,3,4,5,14,6,7,8,15]. Our focus in this review has been mainly on pure evolutionary computing techniques and their hybrids in which power of several methods are combined to produce a superior algorithm for global search.…”

Section: Types Of Optimization Algorithmsmentioning

confidence: 99%

Pure and Hybrid Evolutionary Computing in Global Optimization of Chemical Structures: from Atoms and Molecules to Clusters and Crystals

Sarkar,

Bhattacharyya

2015

Preprint

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The growth of evolutionary computing (EC) methods in the exploration of complex potential energy landscapes of atomic and molecular clusters, as well as crystals over the last decade or so is reviewed. The trend of growth indicates that pure as well as hybrid evolutionary computing techniques in conjunction of DFT has been emerging as a powerful tool, although work on molecular clusters has been rather limited so far. Some attempts to solve the atomic/molecular Schrodinger Equation (SE) directly by genetic algorithms (GA) are available in literature. At the Born-Oppenheimer level of approximation GA-density methods appear to be a viable tool which could be more extensively explored in the coming years, specially in the context of designing molecules and materials with targeted properties.

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Section: Types Of Optimization Algorithmsmentioning

confidence: 99%

Pure and Hybrid Evolutionary Computing in Global Optimization of Chemical Structures: from Atoms and Molecules to Clusters and Crystals

Sarkar,

Bhattacharyya

2015

Preprint

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“…The special case k = 2 was considered in [8], where a bijection between 2-noncrossing trees with a root labelled 2 and 5-ary trees was constructed. The more general case was studied in [6], see also [5]. In analogy to Theorem 1.1, we will also be counting k-noncrossing trees by the number of vertices of each label.…”

Section: Introductionmentioning

confidence: 99%

Refined enumeration of $k$-plane trees and $k$-noncrossing trees

Okoth¹,

Wagner²

2022

Preprint

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A k-plane tree is a plane tree whose vertices are assigned labels between 1 and k in such a way that the sum of the labels along any edge is no greater than k + 1. These trees are known to be related to (k + 1)-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for k-noncrossing trees, a similarly defined family of labelled noncrossing trees that are related to (2k + 1)-ary trees.

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“…These trees were generalized by Pang and Lv in [7] to k-noncrossing trees. A k-noncrossing tree is a noncrossing tree where each node receives a label in {1, 2, .…”

Section: Introductionmentioning

confidence: 99%

Refined enumeration of 2-noncrossing trees

Okoth¹

2021

NNTDM

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A 2-noncrossing tree is a connected graph without cycles that can be drawn in the plane with its vertices on the boundary of circle such that the edges are straight line segments that do not cross and all the vertices are coloured black and white with no ascent (i, j), where i and j are black vertices, in a path from the root. In this paper, we use generating functions to prove a formula that counts 2-noncrossing trees with a black root to take into account the number of white vertices of indegree greater than zero and black vertices. Here, the edges of the 2-noncrossing trees are oriented from a vertex of lower label towards a vertex of higher label. The formula is a refinement of the formula for the number of 2-noncrossing trees that was obtained by Yan and Liu and later on generalized by Pang and Lv. As a consequence of the refinement, we find an equivalent refinement for 2-noncrossing trees with a white root, among other results.

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K-Noncrossing Trees and K-Proper Trees

Cited by 4 publications

References 7 publications

Pure and Hybrid Evolutionary Computing in Global Optimization of Chemical Structures: from Atoms and Molecules to Clusters and Crystals

Pure and Hybrid Evolutionary Computing in Global Optimization of Chemical Structures: from Atoms and Molecules to Clusters and Crystals

Refined enumeration of $k$-plane trees and $k$-noncrossing trees

Refined enumeration of 2-noncrossing trees

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