Binomial Determinants for Tiling Problems Yield to the Holonomic Ansatz
| Autor: | Thotsaporn \\'Aek\\' Thanatipanonda, Elaine Wong, Hao Du, Christoph Koutschan |
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| Rok vydání: | 2021 |
| Předmět: |
Computer Science - Symbolic Computation
FOS: Computer and information sciences Binomial (polynomial) Holonomic 010102 general mathematics Diagonal Rhombus 0102 computer and information sciences Symbolic Computation (cs.SC) Symbolic computation 01 natural sciences Combinatorics symbols.namesake Matrix (mathematics) 010201 computation theory & mathematics Kronecker delta symbols FOS: Mathematics Discrete Mathematics and Combinatorics Mathematics - Combinatorics Computer Science::Symbolic Computation Combinatorics (math.CO) 0101 mathematics Mathematics Ansatz |
| DOI: | 10.48550/arxiv.2105.08539 |
| Popis: | We present and prove closed form expressions for some families of binomial determinants with signed Kronecker deltas that are located along an arbitrary diagonal in the corresponding matrix. They count cyclically symmetric rhombus tilings of hexagonal regions with triangular holes. We extend a previous systematic study of these families, where the locations of the Kronecker deltas depended on an additional parameter, to families with negative Kronecker deltas. By adapting Zeilberger's holonomic ansatz to make it work for our problems, we can take full advantage of computer algebra tools for symbolic summation. This, together with the combinatorial interpretation, allows us to realize some new determinantal relationships. From there, we are able to resolve all remaining open conjectures related to these determinants, including one from 2005 due to Lascoux and Krattenthaler. Comment: 45 pages; Supplementary material at https://wongey.github.io/binom-det |
| Databáze: | OpenAIRE |
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