...D_pn.(1^[l]1)(1^[l]2)...(1^[l]m) = A,
([l] = [lambda])

and the law by which the operation is performed upon the product shows
that the solutions of the given problem are enumerated by the number
A, and that the process of operation actually represents each
solution.

_Ex. Gr._--Take [lambda]1 = 3, [lambda]2 = 2, [lambda]4 = 1,

p1 = 2, p2 = 2, p3 = 1, p4 = 1,

D2^2D1^2.a3a2a1 = 8,

and the process yields the eight diagrams:--

+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
| 1 | 1 | | | 1 | 1 | | | | 1 | 1 | | 1 | 1 | |
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
| 1 | 1 | | | 1 | 1 | | | 1 | 1 | | | | 1 | 1 |
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
| 1 | | | | | | 1 | | 1 | | | | 1 | | |
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
| | | 1 | | 1 | | | | 1 | | | | 1 | | |
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+

+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
| 1 | | 1 | | 1 | | 1 | | 1 | 1 | | | 1 | 1 | |
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
| 1 | 1 | | | 1 | 1 | | | 1 | | 1 | | 1 | | 1 |
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
| 1 | | | | | 1 | | | 1 | | | | | 1 | |
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+
| | 1 | | | 1 | | | | | 1 | | | 1 | | |
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+

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
h_[lambda]1.h_[lambda]2...h_[lambda]m, h_[lambda]m being the
homogeneous product sum of the quantities a, of order [lambda]. The
operator is as before

D_p1.D_p2...D_pn,

and the solutions are enumerated by

D_p1.D_p2...D_pn.h_[lambda]1.h_[lambda]2...h_[lambda]m.

Putting as before [lambda]1 = 2, [la