Studies on reverse plane partition with orthogonal polynomials and integrable systems

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03402
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Osaka Seikei University (2023); Kyoto University (2019-2022)
• Principal Investigator: Kamioka Shuhei 大阪成蹊大学, データサイエンス学部, 准教授 (70543297)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 平面分割 / 組合せ論 / 直交多項式 / 可積分系 / 乱択アルゴリズム

Outline of Final Research Achievements

Reverse plane partitions are combinatorial objects which admit nice generating functions (partition functions) expressible in products. This study aims for discovering new nice generating functions for reverse plane partitions and related combinatorial objects, and for applying the underlying idea to another related problem. Orthogonal polynomials and integrable systems are utilized as tools for that purpose. The main results are the following: (i) Several new nice generating functions for reverse plane partitions and related combinatorial objects are constructed. Those are not special cases of the known ones. (ii) Based on how to construct nice generating functions, a fast algorithm to generate reverse plane partitions at random is developed.

Free Research Field

数え上げ組合せ論

Academic Significance and Societal Importance of the Research Achievements

組合せ論的オブジェクトである逆平面分割は，それそのものに関する研究だけでなく，数学の他分野に関連付けられたり，物理学などの他領域に応用されたりしている．その背景にはきれいな積の形に表せる「よい母関数」の数学的な扱いやすさがある．また，確率論や統計力学とのつながりも深く，そのような研究では逆平面分割をランダムに生成するアルゴリズム（乱択アルゴリズム）が必要になる．本研究の意義は，直交多項式や可積分系など他分野の道具をうまく用いることで，逆平面分割などのまったく新しい母関数を発見した点，および，逆平面分割の高速な乱択アルゴリズムを開発した点にある．