Transfer matrix
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
Two arrows in and two out
An Eulerian circuit enters every point as often as it leaves, so it hands every edge an arrow with the arrows balanced at each point. Count those balanced orientations instead of the circuits and the question turns into the ice in a freezer: on the square grid, where every point has two arrows in and two out, there are about 1.5396 of them per point — Lieb's (4/3)^(3/2), against Pauling's guess of 1.5. Fix the walls of a square of ice and the count becomes 1, 2, 7, 42, 429, 7436: the alternating-sign matrices.
Three arrows in and three out
Put an arrow on every edge of the cubic grid so that every point has three arrows in and three out. On the square grid the same question — two in, two out — has an exact answer, found by Elliott Lieb in 1967: about 1.5396 arrangements per point. On the cubic grid nobody knows the answer. Pauling's guess of 2.5 is a proven lower bound, and exact counts on prisms of the grid come down towards it from above — 2.794, 2.682, 2.587 as the cross-section widens — without reaching it.
Named alongside it
The objects these essays reach for when they reach for this one.
DeterminantEigenvalueEulerian circuitEulerian orientationExhaustive searchGraph colouringLower boundOrientationResidual entropySpanning tree