一类斐波那契相似立方体及其计数性质
A Class of Fibonacci Similar Cubes and Their Counting Properties
DOI: 10.12677/PM.2023.1312381, PDF,  被引量    国家自然科学基金支持
作者: 周玉玉, 陈芳娣*：西北师范大学，数学与统计学院，甘肃 兰州；赵姁姁：甘肃省陕西师范大学平凉实验中学，甘肃 平凉
关键词: 匹配型分配格；滤子格；计数性质；斐波那契相似立方体；秩生成函数；立方体多项式；Matched Distributive Lattices； Filter Lattices； Enumeration Properties； Fibonacci Similar Cubes； Rank Generating Functions； Cube Polynomials
摘要: L-si-Fibonaccene是一个基于Fibonaccene简单变形得到的六角系统，在其内对偶图上建立了一种特殊偏序，命名为L-栅栏。根据偏序L-栅栏的所有滤子按照反包含关系形成的滤子格，得到了一类新的匹配型分配格，忽略掉其Hasse图中的方向后就得到一个结构和性质都与斐波那契立方体相似的新立方体，并计算了它的秩生成函数、立方体多项式和极大立方体多项式.本文的研究不但为匹配型分配格增加了一类新成员，而且其导出的立方体还可以作为新的内联网的模型。
Abstract: L-si-Fibonaccene is a hexagonal system based on the simple deformation of Fibonaccene, and a special partial order is established on its dual graph, which is named L-fence. According to the filter lattice formed by all the filters of the partially ordered L-barrier according to the inverse inclusion relation, a new class of matched distributive lattices is obtained. After ignoring the direction in its Hasse diagram, a new cube with similar structure and properties to the Fibonacci cube is obtained, and its rank generating function, cube polynomial and maximal cube polynomial are calculated. The research in this paper not only adds a new class of members to the matching distributive lattice, but also the derived cube can be used as a new model of the intranet.
文章引用：周玉玉, 陈芳娣, 赵姁姁. 一类斐波那契相似立方体及其计数性质[J]. 理论数学, 2023, 13(12): 3676-3689. https://doi.org/10.12677/PM.2023.1312381

参考文献

[1] Hsu, W.J. (1993) Fibonacci Cubes—A New Interconnection Topology. IEEE Transactions on Parallel and Distributed Systems, 4, 3-12. [Google Scholar] [CrossRef] 
[2] Hsu, W.J., Carl, V.P. and Liu, J.S. (1993) Fibonacci Cubes—A Class of Self-Similar Graphs. The Fibonacci Quarterly, 31, 65-72.
[3] Munarini, E., Cippo, C.P. and Zagaglia Salvi, N. (2001) On the Lucas Cubes. The Fibonacci Quarterly, 39, 12-21.
[4] Zagaglia Salvi, N. (2001) The Lucas Lattice. The 2001 International Parallel and Distributed Processing Symposium, Vol. 2, San Francisco, 23-27 April 2001, 719-721.
[5] Munarini, E. and Zagaglia Salvi, N. (2002) On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns. Discrete Mathematics, 259, 163-177. [Google Scholar] [CrossRef] 
[6] Klavžar, S. and Mollard, M. (2012) Cube Polynomial of Fibonacci and Lucas Cubes. Acta Applicandae Mathematicae, 117, 93-105. [Google Scholar] [CrossRef] 
[7] Saygı, E. and Eğecioğlu, Ö. (2018) q-Counting Hypercubes in Lucas Cubes. Turkish Journal of Mathematics, 42, 190-203. [Google Scholar] [CrossRef] 
[8] Mollard, M. (2012) Maximal Hypercubes in Fibonacci and Lucas Cubes. Discrete Applied Mathematics, 160, 2479-2483. [Google Scholar] [CrossRef] 
[9] Gravier, S., Mollard, M., Špacapaand, S. and Zemljič, S.S. (2015) On Disjoint Hypercubes in Fibonacci Cubes. Discrete Applied Mathematics, 190-191, 50-55. [Google Scholar] [CrossRef] 
[10] Saygı, E. and Eğecioğlu, O. (2016) Counting Disjoint Hypercubes in Fibonacci Cubes. Discrete Applied Mathematics, 215, 231-237. [Google Scholar] [CrossRef] 
[11] Klavžar, S., Mollard, M. and Petkovšek, M. (2011) The Degree Sequence of Fibonacci and Lucas Cubes. Discrete Mathematics, 311, 1310-1322. [Google Scholar] [CrossRef] 
[12] Klavžar, S. (2013) Structure of Fibonacci Cubes: A Survey. Journal of Combinatorial Optimization, 25, 505-522. [Google Scholar] [CrossRef] 
[13] Lam, P.C.B. and Zhang, H. (2003) A Distributive Lattice on the Set of Perfect Matchings of a Plane Bipartite Graph. Order, 20, 13-29. [Google Scholar] [CrossRef] 
[14] Zhang, H., Yang, D. and Yao, H. (2014) Decomposition Theorem on Matchable Distributive Lattices. Discrete Applied Mathematics, 166, 239-248. [Google Scholar] [CrossRef] 
[15] Klavžar, S. and Žigert Pleteršek, P. (2005) Fibonacci Cubes Are the Resonance Graphs of Fibonaccenes. The Fibonacci Quarterly, 43, 269-276.
[16] Yao, H. and Zhang, H. (2015) Non-Matchable Distributive Lattices. Discrete Mathematics, 338, 122-132. [Google Scholar] [CrossRef] 
[17] Zhang, H., Ouand, L. and Yao, H. (2009) Fibonacci-Like Cubes as Z-Transformation Graphs. Discrete Mathematics, 309, 1284-1293. [Google Scholar] [CrossRef] 
[18] Davey, B.A. and Priestley, H.A. (2002) Introduction to Lattices and Order. 2nd Edition, Cambridge University Press, Cambridge. [Google Scholar] [CrossRef] 
[19] Stanley, R.P. (2011) Enumerative Combinatorics: Volume 1, Volume 49 of Cambridge Studies in Advanced Mathematics. 2nd Edition, Cambridge University Press, Cam-bridge.
[20] Wang, X., Zhao, X. and Yao, H. (2018) Convex Expansion for Finite Distributive Lattices with Applica-tions. arXiv: 1810. 06762.
[21] Zhang, H. (2006) Z-Transformation Graphs of Perfect Matchings of Plane Bipartite Graphs: A Survey. MATCH Communications in Mathematical and in Computer Chemistry, 56, 457-476.
[22] Comtet, L. (1974) Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. Reidel Publishing Company, Dor-drecht.
[23] Sloane, N.J.A. (2019) The On-Line Encyclopedia of Integer Sequences. http://oeis.org/ [Google Scholar] [CrossRef] 

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