Hafnian of two-parameter matrices

[2101.09722] Hafnian of two-parameter matrices Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2101.09722 (math) [Submitted on 24 Jan 2021] Title: Hafnian of two-parameter matrices Authors: Dmitry Efimov View a PDF of the paper titled Hafnian of two-parameter matrices, by Dmitry Efimov View PDF HTML (experimental) Abstract: The concept of the hafnian first appeared in the works on quantum field theory by E. R. Caianiello. However, it also has an important combinatorial property: the hafnian of the adjacency matrix of an undirected weighted graph is equal to the total sum of the weights of perfect matchings in this graph. In general, the use of the hafnian is limited by the complexity of its computation. In this paper, we present an efficient method for the exact calculation of the hafnian of two-parameter matrices. In terms of graphs, we count the total sum of the weights of perfect matchings in graphs whose edge weights take only two values. This method is based on the formula expressing the hafnian of a sum of two matrices through the product of the hafnians of their submatrices. The necessary condition for the application of this method is the possibility to count the number of k-edge matchings in some graphs. We consider two special cases in detail using a Toeplitz matrix as the two-parameter matrix. As an example, we propose a new interpretation of some of the sequences from the On-Line Encyclopedia of Integer Sequences and then provide new analytical formulas to count the number of certain linear chord diagrams. Comments: 12 pages, 8 figures Subjects: Combinatorics (math.CO) MSC classes: 15A15, 05C30, 05A15 (Primary), 15B05 (Secondary) Cite as: arXiv:2101.09722 [math.CO] (or arXiv:2101.09722v1 [math.CO] for this version) https://doi.org/10.48550/arXiv.2101.09722 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Proceedings of the Komi Science Centre of the Ural Division of the Russian Academy of Sciences. 2022. N 5(57). pp. 15-19 Related DOI : https://doi.org/10.19110/1994-5655-2022-5-15-19 Focus to learn more DOI(s) linking to related resources Submission history From: Dmitry Efimov [ view email ] [v1] Sun, 24 Jan 2021 14:32:21 UTC (10 KB) Full-text links: Access Paper: View a PDF of the paper titled Hafnian of two-parameter matrices, by Dmitry Efimov View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2021-01 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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