Lieb's square ice constant

Lieb's square ice constant is a mathematical constant used in the field of combinatorics to quantify the number of Eulerian orientations of grid graphs. It was introduced by Elliott H. Lieb in 1967.[1]

Binary 1.10001010001000110100010111001100…
Decimal 1.53960071783900203869106341467188…
Hexadecimal 1.8A2345CC04425BC2CBF57DB94EDCA6B2…
Continued fraction
Algebraic form

Definition

An n × n grid graph (with periodic boundary conditions and n ≥ 2) has n2 vertices and 2n2 edges; it is 4-regular, meaning that each vertex has exactly four neighbors. An orientation of this graph is an assignment of a direction to each edge; it is an Eulerian orientation if it gives each vertex exactly two incoming edges and exactly two outgoing edges.

Denote the number of Eulerian orientations of this graph by f(n). Then

[2]

is Lieb's square ice constant. Lieb used a transfer-matrix method to compute this exactly.

The function f(n) also counts the number of 3-colorings of grid graphs, the number of nowhere-zero 3-flows in 4-regular graphs, and the number of local flat foldings of the Miura fold.[3] Some historical and physical background can be found in the article Ice-type model.

See also

References
  1. Lieb, Elliott (1967). "Residual Entropy of Square Ice". Physical Review. 162 (1): 162. doi:10.1103/PhysRev.162.162.
  2. (sequence A118273 in the OEIS)
  3. Ballinger, Brad; Damian, Mirela; Eppstein, David; Flatland, Robin; Ginepro, Jessica; Hull, Thomas (2015), "Minimum forcing sets for Miura folding patterns", Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, pp. 136–147, arXiv:1410.2231, doi:10.1137/1.9781611973730.11
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