viz. every solution of the problem. Observe that transposition of the diagrams furnishes a proof of the simplest of the laws of symmetry in the theory of symmetric functions.
For the next example we have a similar problem, but no restriction is placed upon the magnitude of the numbers which may appear in the compartments. The function is now hλ1hλ2 ... hλm, hλm being the homogeneous product sum of the quantities a, of order λ. The operator is as before
Dp1Dp2 ... Dpn,
and the solutions are enumerated by
Dp1Dp2 ... Dpn hλ1hλ2 ... hλm.
Putting as before λ1 = 2, λ2 = 2, λ3 = 1, p1 = 2, p2 = 2, p3 = 1, p4 = 1, the reader will have no difficulty in constructing the diagrams of the eighteen solutions.
The next and last example of a multitude that might be given shows the extraordinary power of the method by solving the famous problem of the “Latin Square,” which for hundreds of years had proved beyond the powers of mathematicians. The problem consists in placing n letters a, b, c, ... n in the compartments of a square lattice of n² compartments, no compartment being empty, so that no letter occurs twice either in the same row or in the same column. The function is here
(Σ α12 n-1 α22 n-2 ... α2n-1 αn)n,
and the operator Dn2 n-1, the enumeration being given by