Solving Tiling Enumeration Problems by Tensor Network Contractions
Abstract
This paper presents an algorithm for contracting two-dimensional tensor networks on a square lattice; and we combine it with solving congruence equations to compute the exact enumeration (including weighted enumeration) of Wang tilings. Based on this, the paper demonstrates how to transform other tiling enumeration problems (such as those of polyominoes) into Wang tiling enumeration problems, thereby solving them using this algorithm. Our algorithm extends the sequence length records for dozens of sequences defined by polyomino tiling enumeration on chessboards on the OEIS website, covering numerous of different polyomino sets, including I-polyominoes, tetrominoes, pentominoes, etc. This demonstrates the high efficiency and strong universality of the algorithm for solving exact tiling enumeration problems. In addition, the theory and techniques used in the algorithm establish a bridge between tensor network contractions and tiling enumeration, where the former provides a theoretical foundation for solving problems in the latter, while the latter offers an intuitive combinatorial interpretation of the former.
Related
- Extended Generalized Contractions, Metric Completeness, and Applications to Solving Nonlinear Diophantine Equations
- Relational nonlinear almost contractions of Lakshmikantham-Ćirić type and applications to boundary value problems
- Intractable enumeration problems are like Russian nesting dolls: structural properties of monomer-dimer coverings on two-dimensional quadratic lattices
- Solving Inverse Dirac-weighted Sturm-Liouville Problems via Cauchy problems
- Solving Multiobjective Linear Set Optimization Problems
BibTeX
@article{liang2026solving,
title = {{Solving Tiling Enumeration Problems by Tensor Network Contractions}},
author = {Kai Liang},
journal = {The Electronic Journal of Combinatorics},
year = {2026},
month = nov,
doi = {10.37236/14510},
url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p33},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Solving Tiling Enumeration Problems by Tensor Network Contractions
AU - Kai Liang
PY - 2026
DA - 2026/11/06
JO - The Electronic Journal of Combinatorics
DO - 10.37236/14510
UR - https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p33
AB - This paper presents an algorithm for contracting two-dimensional tensor networks on a square lattice; and we combine it with solving congruence equations to compute the exact enumeration (including weighted enumeration) of Wang tilings. Based on this, the paper demonstrates how to transform other tiling enumeration problems (such as those of polyominoes) into Wang tiling enumeration problems, thereby solving them using this algorithm. Our algorithm extends the sequence length records for dozens of sequences defined by polyomino tiling enumeration on chessboards on the OEIS website, covering numerous of different polyomino sets, including I-polyominoes, tetrominoes, pentominoes, etc. This demonstrates the high efficiency and strong universality of the algorithm for solving exact tiling enumeration problems. In addition, the theory and techniques used in the algorithm establish a bridge between tensor network contractions and tiling enumeration, where the former provides a theoretical foundation for solving problems in the latter, while the latter offers an intuitive combinatorial interpretation of the former.
SN - 1077-8926
ER -