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Research on algebraic combinatorics related to matrices and hypergeometric series and surrounding topics

Research Project

Project/Area Number 16K05060
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionWakayama University

Principal Investigator

Tagawa Hiroyuki  和歌山大学, 教育学部, 教授 (80283943)

Project Period (FY) 2016-04-01 – 2021-03-31
Project Status Completed (Fiscal Year 2020)
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)
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
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.

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。

Report

(6 results)
  • 2020 Annual Research Report   Final Research Report ( PDF )
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (8 results)

All 2021 2020 2019 2018 Other

All Journal Article (4 results) (of which Open Access: 4 results) Remarks (4 results)

  • [Journal Article] On convolutions of extended Narayana polynomials2021

    • Author(s)
      田川裕之
    • Journal Title

      和歌山大学教育学部紀要. 自然科学

      Volume: 71 Pages: 135-141

    • DOI

      10.19002/AN00257977.71.135

    • NAID

      120006959990

    • ISSN
      13424645
    • URL

      https://wakayama-u.repo.nii.ac.jp/records/2003323

    • Year and Date
      2021-02-08
    • Related Report
      2020 Annual Research Report
    • Open Access
  • [Journal Article] On convolutions of extended Catalan numbers2020

    • Author(s)
      田川裕之
    • Journal Title

      和歌山大学教育学部紀要. 自然科学

      Volume: 70 Pages: 11-15

    • DOI

      10.19002/AN00257977.70.11

    • NAID

      120006811607

    • ISSN
      13424645
    • URL

      https://wakayama-u.repo.nii.ac.jp/records/2003310

    • Year and Date
      2020-02-04
    • Related Report
      2019 Research-status Report
    • Open Access
  • [Journal Article] An investigation of Fibonacci numbers as a teaching material of mathematics education and an extension of the formula of the power sum of Fibonacci numbers2019

    • Author(s)
      北山秀隆, 松山ともこ, 西口正純, 西山尚志, 田川裕之, 田窪佳寿子
    • Journal Title

      和歌山大学教育学部紀要. 教育科学

      Volume: 69 Pages: 59-66

    • DOI

      10.19002/AN00257966.69.59

    • NAID

      120006560949

    • ISSN
      13425331
    • URL

      https://wakayama-u.repo.nii.ac.jp/records/2003059

    • Year and Date
      2019-02-08
    • Related Report
      2018 Research-status Report
    • Open Access
  • [Journal Article] An Investigation and Extension of Tower of Hanoi as a Teaching Material of Mathematics Education2018

    • Author(s)
      北山秀隆, 南拓弥, 西山尚志, 田川裕之, 鷲山峻大, 山本紀代
    • Journal Title

      和歌山大学教育学部紀要. 教育科学

      Volume: 68 Issue: 1 Pages: 189-196

    • DOI

      10.19002/AN00257966.68(1).189

    • NAID

      120006415590

    • ISSN
      13425331
    • URL

      https://wakayama-u.repo.nii.ac.jp/records/2003013

    • Year and Date
      2018-01-31
    • Related Report
      2017 Research-status Report
    • Open Access
  • [Remarks] The home page of Hiroyuki TAGAWA

    • URL

      http://web.wakayama-u.ac.jp/~tagawa/

    • Related Report
      2020 Annual Research Report
  • [Remarks] 和歌山大学教育学部田川研究室ホームページ

    • URL

      http://web.wakayama-u.ac.jp/~tagawa/

    • Related Report
      2019 Research-status Report 2018 Research-status Report
  • [Remarks] 和歌山大学教育学部田川研究室ホームページ

    • URL

      http://www.center.wakayama-u.ac.jp/~tagawa/

    • Related Report
      2017 Research-status Report
  • [Remarks] 和歌山大学教育学部田川研究室ホームページ

    • URL

      http://www.wakayama-u.ac.jp/~tagawa

    • Related Report
      2016 Research-status Report

URL: 

Published: 2016-04-21   Modified: 2022-01-27  

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