Binomial determinants for tiling problems yield to the holonomic ansatz

Hao Du; Christoph Koutschan; Thotsaporn Thanatipanonda; Elaine Wong

doi:10.1016/j.ejc.2021.103437

Binomial determinants for tiling problems yield to the holonomic ansatz

Hao Du
, Christoph Koutschan
, Thotsaporn Thanatipanonda
, Elaine Wong

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

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.

Original language	English
Article number	103437
Journal	European Journal of Combinatorics
Volume	99
DOIs	https://doi.org/10.1016/j.ejc.2021.103437
State	Published - Jan 2022
Externally published	Yes

Funding

C. Koutschan and E. Wong are both supported by the Austrian Science Fund (FWF): F5011-N15 . H. Du is supported by the Austrian Science Fund (FWF): F5011-N15 , P31952 , and P32301 . These three authors would like to acknowledge the Special Research Program “Algorithmic and Enumerative Combinatorics” that encouraged and supported this research. The authors would also like to thank Mihai Ciucu and Ilse Fischer for the update regarding Conjecture 39 from [21] . E. Wong would like to thank Manfred Buchacher, Matteo Gallet and Ali Uncu for their friendship, support and advice during the preparation of this manuscript, the Symbolic Computation group at RICAM who gave ample freedom and space to conduct research, and especially RICAM’s system administrators Florian Tischler and Wolfgang Forsthuber who provided technical support and infrastructure for the computations in this paper to complete. We would also like to thank the referees for their careful reading and helpful comments.

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title = "Binomial determinants for tiling problems yield to the holonomic ansatz",

abstract = "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.",

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AB - 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.

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Binomial determinants for tiling problems yield to the holonomic ansatz

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