Project maintained by wongey Hosted on GitHub Pages — Theme by mattgraham

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.

- Hao Du, Beijing University of Posts and Telecommunications
- Christoph Koutschan, Austrian Academy of Sciences, RICAM
- Thotsaporn Thanatipanonda, Mahidol University International College, Thailand
- Elaine Wong*, Austrian Academy of Sciences, RICAM

*Corresponding Author

- June 17, 2021 at 13:45 (Elaine Wong; Algebra and Discrete Mathematics Seminar, Johannes Kepler University, Online)
- July 2, 2021 at 10:30 (Christoph Koutschan; Combinatorics Seminar, Lyon à l’ENS, Online)
- July 23-27, 2021 time TBD (Elaine Wong; International Conference on Applications of Computer Algebra, Online)
- July 26-29, 2021 time TBD (Christoph Koutschan; Combinatorics and Algebras From {A}mitav to {Z}eilberger Workshop, Online)
- August 16-20, 2021 time TBD (Elaine Wong; SIAM Conference on Applied Algebraic Geometry, Online)

This paper has been submitted for publication.

Preprints can be found on arXiv:2105.08539 and as a RICAM Report.

- The following files contain a compilation of experiments that check identities in the statements of key lemmas, propositions, theorems and corollaries in our paper by instantiating for small values of r and m. The results of our experiments are compiled here as both a Mathematica notebook (with a pdf version for those without a Mathematica installation) and a Maple script.
- Sanity Check (Mathematica): Download zip file (63 KB).
- Sanity Check (Maple): Download text file (11 KB).

- The following Mathematica notebooks require the Mathematica package HolonomicFunctions.
- Proof of Lemma 10: Eq1 Zip (32 MB), Eq2 Zip (59 MB)
- Proof of Lemma 16: Eq3 Zip (667 KB), Eq4 Zip (669 KB)
- Proof of Lemma 17: Eq5 Zip (711 KB)