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.

Authors

• Hao Du, Beijing University of Posts and Telecommunications

  • Contact: [email protected]
  • Previous Affiliation: Johannes Kepler University, Linz, Austria

• Christoph Koutschan, Austrian Academy of Sciences, RICAM

  • Contact: [email protected]

• Thotsaporn Thanatipanonda, Mahidol University International College, Thailand

  • Contact: [email protected]

• Elaine Wong, Oak Ridge National Laboratory, USA

  • Contact (Corresponding Author): [email protected]
  • Previous Affiliation: Austrian Academy of Sciences, RICAM

Paper

• This paper has been published in the European Journal of Combinatorics with DOI:10.1016/j.ejc.2021.103437.

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

• Cite our paper by using this bib.

Files

• 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 16 (alt): Eq3 Alt Zip (532 KB), Eq4 Alt Zip (534 KB)
  • Proof of Lemma 17: Eq5 Zip (711 KB)

Past and Upcoming Talks

All times are CEST.

• October 21, 2021 at 19:00, Online
  • Elaine Wong; Algebraic and Enumerative Combinatorics Seminar, University of Waterloo
  • [Slides]
• September 27, 2021 at 17:30, Online, In German
  • Christoph Koutschan; DMV-ÖMG Jahrestagung 2021, Computer Algebra, University of Passau
  • [Slides]
• August 16, 2021 at 21:10, Online
  • Elaine Wong; SIAM Conference on Applied Algebraic Geometry, Symbolic Combinatorics
  • [Slides]
• August 10, 2021 at 10:30, In Person
  • Elaine Wong; AEC SFB Statusseminar, Johannes Kepler University
  • [Slides]
• July 29, 2021 at 16:00, Online
  • Christoph Koutschan; Combinatorics and Algebras From {A}mitav to {Z}eilberger Workshop
  • [Slides]
• July 24, 2021 at 17:30, Online
  • Elaine Wong; International Conference on Applications of Computer Algebra, Algorithmic Combinatorics
  • [Slides]
• July 2, 2021 at 10:30, Online
  • Christoph Koutschan; Combinatorics Seminar, Lyon à l'ENS
  • [Slides]
• June 17, 2021 at 13:45, Online
  • Elaine Wong; Algebra and Discrete Mathematics Seminar, Johannes Kepler University
  • [Slides]