Twenty-Vertex Entropy Constant

Let  denote the  triangular lattice with wraparound. Call an orientation of  an assignment of a direction to each edge of , and denote the number of orientations of  such that each vertex has three inwardly directed and three outwardly directly edges by . Such an orientation is said to consist of Eulerian orientation.

Baxter showed that

			(1)
			(2)
			(3)

(OEIS A104956; Finch 2003, p. 412), which is known as the twenty-vertex entropy constant (a term coined here for the first time).

REFERENCES:

Baxter, R. J. " Model on a Triangular Lattice." J. Math. Phys. 10, 1211-1216, 1969.

Baxter, R. J. Exactly Solved Models in Statistical Mechanics. New York: Academic Press, 1982.

Finch, S. R. "Lieb's Square Ice Constant." §5.24 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 412-413, 2003.

Sloane, N. J. A. Sequence A104956 in "The On-Line Encyclopedia of Integer Sequences."