2023 Fiscal Year Final Research Report
A study of combinatorial problems caused by the crossing of chords
| Project/Area Number |
19K03607
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12030:Basic mathematics-related
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| Research Institution | Shonan Institute of Technology |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | コードダイアグラム / 三角形分割 / ヤング図形 / 増大木 / 幾何グラフ |
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| Outline of Final Research Achievements |
A chord diagram is a set of chords having no common endpoints. In this study, we focus on chord expansion, the operation of generating two new chord diagrams for a given chord diagram E by resolving the crossings of the chords contained in the diagram.
This operation is repeated one after the other until finally chord diagrams with no crossings are generated. The resulting multiset of nonintersecting code diagrams, NCD(E), is uniquely determined regardless of the order of the chord expansion. In this study, the properties of NCD(E) are investigated in detail from a combinatorial viewpoint. In particular, we show that the cardinality of NCD(E) coincides with that of other combinatorial structures, such as certain Young diagrams and increasing trees.
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| Free Research Field |
離散数学
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| Academic Significance and Societal Importance of the Research Achievements |
本研究の学術的意義は，平面上の１つの円における複数の弦の配置、という古代より人類が親しんできた対象について新たな知見を付け加えたことである．この弦の配置について交差の展開操作を繰り返して施すことにより結果的にそれぞれの配置が交差を含まない配置にまで還元される．それに関連する数え上げ問題において、他の重要な組み合わせ的な構造、交代置換、増大木、および0-1ヤング図形など、と密接な関連があることが明らかになった．今後の方向としては、単に数え上げ問題だけでなく弦の配置の構造と他の組合せ構造との間の1対1対応を示すことができれば、さらに弦の配置の研究の重要性を補強することになると思われる．
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