A Curious Family of Binomial Determinants That Count Rhombus Tilings of a Holey Hexagon
Christoph Koutschan and Thotsaporn "Aek" Thanatipanonda
AbstractWe evaluate a curious determinant, first mentioned by George Andrews in 1980 in the context of descending plane partitions. Our strategy is to combine the famous Desnanot-Jacobi-Carroll identity with automated proof techniques. More precisely, we follow the holonomic ansatz that was proposed by Doron Zeilberger in 2007. We derive a compact and nice formula for Andrews's determinant, and use it to solve a challenge problem that we posed in a previous paper. By noting that Andrew's determinant is a special case of a two-parameter family of determinants, we find closed forms for several one-parameter subfamilies. The interest in these determinants arises because they count cyclically symmetric rhombus tilings of a hexagon with several triangular holes inside.
PaperThe material on this webpage accompanies the article A Curious Family of Binomial Determinants That Count Rhombus Tilings of a Holey Hexagon by Christoph Koutschan and Thotsaporn "Aek" Thanatipanonda.
Supplementary electronic materialWe provide a Mathematica notebook
- CuriousDeterminants.nb (Mathematica notebook)
- CuriousDeterminants.cdf (Mathematica computable document)
- CuriousDeterminants.pdf (pdf printout of the notebook)
Additionally, we provide some precomputed results, namely the recursive descriptions of the auxiliary functions cn,j that appear in the proofs, and the corresponding creative telescoping relations (although their computation doesn't require an excessive amount of time). Just unpack the zip file into the same directory where the notebook is stored.
- precomputed.zip (1.4 MB)