Mathematical constant used in combinatorics
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.
Definition.
An "n" × "n" grid graph (with periodic boundary conditions and "n" ≥ 2) has "n"2 vertices and 2"n"2 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
 formula_1
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. Some historical and physical background can be found in the article Ice-type model.
References.
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