Reserch on enumeration of alternating sign matrices, plane partitions and related determinants and Pfaffians

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400018
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: University of the Ryukyus
• Principal Investigator: Ishikawa Masao 琉球大学, 教育学部, 教授 (40243373)
• Co-Investigator(Kenkyū-buntansha): OKADA Soichi 名古屋大学, 多元数理科学研究科, 教授 (20224016)
• Co-Investigator(Renkei-kenkyūsha): TAGAWA Hiroyuki 和歌山大学, 教育学部, 教授 (80283943)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 数え上げ組合せ論 / 表現論 / 数理物理 / タイリング

Outline of Final Research Achievements

We studied Pfaffian analogue of q-Catalan Hankel determinant and the Askey-Habsieger-Kadell q-Selberg's integral formula using de Bruijn's formula. We also obtained similar results for certain q-Pfaffian whose entries are moments of Al-Salam-Carlitz polynomials using a result by Baker and Forrester. We also studied a q-analogue of Mehta-Wang determinant and reduced it to a quadratic equation for the Askey-Wilson polynomials, which was a joint-work with V. Guo, H. Tagawa and J. Zeng. We also studied a combinatorial proof of a couple of Pfaffian identities with T. Eisenkoelbl and J. Kim. Recently I had some results on the enumeration of domino tilings of Aztec rectangle with holes. The enumeration problems are closely related to the combinatorial methods, e.g. the Gessel-Viennot Lattice path method, Kasteleyn's method and the enumeration of spanning trees of graph. In this research we obtain a determinant expression for the enumeration appealing to the Gessel-Viennot method.

Free Research Field

代数的組合せ論