We exploit the permutation creation ability of genetic optimization to find the permutation of one point set that puts it into correspondence with another one. To this end, we provide a genetic ...algorithm for the 3D shape correspondence problem, which is the main contribution of this article. As another significant contribution, we present an adaptive sampling approach that relocates the matched points based on the currently available correspondence via an alternating optimization. The point sets to be matched are sampled from two isometric (or nearly isometric) shapes. The sparse one-to-one correspondence, i.e., bijection, that we produce is validated both in terms of running time and accuracy in a comprehensive test suite that includes four standard shape benchmarks and state-of-the-art techniques.
Building on a bijection of Vandervelde, we enumerate certain unimodal sequences whose alternating sum equals zero. This enables us to refine the enumeration of strict partitions with respect to the ...number of parts and the BG-rank.
Bijective enumeration of general stacks Guo, Qianghui; Jin, Yinglie; Sun, Lisa H. ...
Advances in applied mathematics,
July 2024, 2024-07-00, Letnik:
158
Journal Article
Recenzirano
Odprti dostop
Combinatorial enumeration of various RNA secondary structures and protein contact maps is of great interest for both combinatorists and computational biologists. Counting protein contact maps is much ...more difficult than that of RNA secondary structures due to the significant higher vertex degree. The state of art upper bound for vertex degree in previous works is two. This paper proposes a solution for counting general stacks with arbitrary vertex degree upper bound. By establishing a bijection between such general stacks and m-regular Λ-avoiding DLU-paths, and counting these pattern avoiding lattice paths, we obtain a unified system of equations for the generating functions of the number of general stacks. We further show that previous enumeration results for RNA secondary structures and linear stacks of protein contact maps can be derived from the equations for general stacks as special cases.
A
d-angulation is a planar map with faces of degree
d. We present for each integer
d
⩾
3
a bijection between the class of
d-angulations of girth
d (i.e., with no cycle of length less than
d) and a ...class of decorated plane trees. Each of the bijections is obtained by specializing a “master bijection” which extends an earlier construction of the first author. Our construction unifies known bijections by Fusy, Poulalhon and Schaeffer for triangulations (
d
=
3
) and by Schaeffer for quadrangulations (
d
=
4
). For
d
⩾
5
, both the bijections and the enumerative results are new.
We also extend our bijections so as to enumerate
p-gonal d-angulations (
d-angulations with a simple boundary of length
p) of girth
d. We thereby recover bijectively the results of Brown for simple
p-gonal triangulations and simple 2
p-gonal quadrangulations and establish new results for
d
⩾
5
.
A key ingredient in our proofs is a class of orientations characterizing
d-angulations of girth
d. Earlier results by Schnyder and by De Fraysseix and Ossona de Mendez showed that simple triangulations and simple quadrangulations are characterized by the existence of orientations having respectively indegree 3 and 2 at each inner vertex. We extend this characterization by showing that a
d-angulation has girth
d if and only if the graph obtained by duplicating each edge
d
−
2
times admits an orientation having indegree
d at each inner vertex.
In 2012 Bóna showed the rather surprising fact that the cumulative number of occurrences of the classical patterns 231 and 213 is the same on the set of permutations avoiding 132, even though the ...pattern based statistics 231 and 213 do not have the same distribution on this set. Here we show that if it is required for the symbols playing the role of 1 and 3 in the occurrences of 231 and 213 to be adjacent, then the obtained statistics are equidistributed on the set of 132-avoiding permutations. Actually, expressed in terms of vincular patterns, we prove bijectively the following more general results: the statistics based on the patterns ▪, ▪ and ▪, together with other statistics, have the same joint distribution on Sn(132), and so do the patterns ▪ and ▪; and up to trivial transformations, these statistics are the only based on length-three proper (not classical nor consecutive) vincular patterns which are equidistributed on a set of permutations avoiding a classical length-three pattern.
Volume-Enhanced Compatible Remeshing of 3D Models Yang, Yang; Fu, Xiao-Ming; Chai, Shuangming ...
IEEE transactions on visualization and computer graphics,
2019-Oct.-1, 2019-Oct, 2019-10-1, 20191001, Letnik:
25, Številka:
10
Journal Article
Recenzirano
Compatible remeshing provides meshes with common connectivity structures. The existing compatible remeshing methods usually suffer from high computational cost or poor quality. In this paper, we ...present a fast method for computing compatible meshes with high quality. Given two closed, oriented, and topologically equivalent surfaces and a sparse set of corresponding landmarks, we first compute a bijective inter-surface mapping, from which compatible meshes are generated. We then improve the remeshing quality by using a volume-enhanced optimization. In contrast to previous work, our method designs a fast volume-enhanced improvement procedure that directly reduces the isometric distortion of the map between the compatible meshes. Our method also tries to preserve the shapes of the input meshes by projecting the vertices of the compatible meshes onto the input surfaces. Central to this approach is the use of the monotone preconditioned conjugate gradient method, which minimizes the energies effectively and efficiently. Compared with state-of-the-art methods, our method performs about one order of magnitude faster with better remeshing quality. We demonstrate the efficiency and efficacy of our method using various model pairs.
As a classical object, the Tamari lattice has many generalizations, including ν-Tamari lattices and parabolic Tamari lattices. In this article, we unify these generalizations in a bijective fashion. ...We first prove that parabolic Tamari lattices are isomorphic to ν-Tamari lattices for bounce paths ν. We then introduce a new combinatorial object called “left-aligned colorable tree”, and show that it provides a bijective bridge between various parabolic Catalan objects and certain nested pairs of Dyck paths. As a consequence, we prove the Steep-Bounce Conjecture using a generalization of the famous zeta map in q,t-Catalan combinatorics. A generalization of the zeta map on parking functions, which arises in the theory of diagonal harmonics, is also obtained as a labeled version of our bijection.
This paper aims to establish two bijections: from regions of the r-Shi arrangement to O-rooted labeled r-trees, and from regions of the r-Catalan arrangement to pairings of permutation and r-Dyck ...path. To this end, we introduce a cubic matrix for each region of the hyperplane arrangements. The first bijection is established by reading the positions of minimal positive entries in its column slices. The second one is obtained by reading the numbers of positive entries in its column slice, which turns out to be essentially the same as the bijection obtained by Duarte and Guedes de Oliveira 10 in 2019. Moreover, by reading the numbers of positive entries in its row slices we will recover the Pak-Stanley labeling, which is a celebrated bijection from regions of the r-Shi arrangement to r-parking functions.
Necklaces and slimes Oh, Suho; Park, Jina
Discrete mathematics,
August 2020, 2020-08-00, Letnik:
343, Številka:
8
Journal Article
Recenzirano
We show there is a bijection between the binary necklaces with n black beads and k white beads and certain (n,k)-codes when n is prime. The main idea is to come up with a new map on necklaces called ...slime migration.
PDF
