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

2020 Fiscal Year Final Research Report

• Project/Area Number: 16K05060
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Wakayama 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
• Keywords: 行列 / 超幾何級数 / Aztec rectangle / hook length formula

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.

Free Research Field

代数的組み合わせ論

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。