Research on algebraic combinatorics related to matrices and hypergeometric series and surrounding topics
| Project/Area Number |
16K05060
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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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| Research Field |
Algebra
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| Research Institution | Wakayama University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2021-03-31
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| Project Status |
Completed (Fiscal Year 2020)
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| Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | 行列 / 超幾何級数 / Aztec rectangle / hook length formula / Aztec diamond / hook formula / フィボナッチ数 / 代数的組合せ論 / 表現論的組合せ論 / 組合せ論的表現論 |
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| Outline of Final Research Achievements |
In this research, we mainly obtained the following results: We expressed the generating function of the domino tilings in the Aztec rectangle with connected holes by a determinant of the matrix whose elements are hypergeometric series. We proved that the number of the Schroder paths which restricted the height was expressed by a summation of some hypergeometric series. We obtained almost same results on the Delannoy pathes. We proved positively a part of Toyosawa conjecture on the hook formula of cylindric skew diagrams. We got many equalities, contiguous relations, summation formulas, product formulas and so on for hypergeometric series through an analysis of extended Narayana polynomials, extended Catalan numbers, extended Fibonacci numbers, extended towers of Hanoi and so on.
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| Academic Significance and Societal Importance of the Research Achievements |
連続して穴の開いた Aztec rectangle の母関数、及び高さを制限した Schroder path, Delannoy path の個数に関して得られた結果は、他の tiling 問題への拡張が見込める。また、本研究課題で得られた多数の超幾何級数の等式、隣接関係式、和公式、積公式等については、直交多項式や表現論などの他分野への応用、類似した等式の発見、ｑ超幾何級数への拡張等が期待できる。さらに、cylindric skew diagram の hook formula に関する結果は、豊澤予想の解決に寄与するものと考えられる。
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Report
(6 results)
Research Products
(8 results)
