the term 5(21²) indicates that objects such as a, a, b, c can be partitioned in five ways into two parts. These are a | a, b, c; b | a; a, c; c | a, a, b; a, a | b, c; a, b | a, c. The function hrs has been studied. (See MacMahon, Proc. Lond. Math. Soc. vol. xix.) Putting x equal to unity, the function may be written (h2 + h4 + h6 + ...) (1 + h1 + h2 + h3 + h4 + ...), a convenient formula.
The method of differential operators, of wide application to problems of combinatorial analysis, has for its leading idea the designing of a function and of a differential operator, Method of differential operators. so that when the operator is performed upon the function a number is reached which enumerates the solutions of the given problem. Generally speaking, the problems considered are such as are connected with lattices, or as it is possible to connect with lattices.
To take the simplest possible example, consider the problem of finding the number of permutations of n different letters. The function is here xn, and the operator (d/dx)n = δnx , yielding δn x xn = n! the number which enumerates the permutations. In fact—
δxxn = δx . x . x . x . x . x . x . ...,
and differentiating we obtain a sum of n terms by striking out an x from the product in all possible ways. Fixing upon any one of these terms, say x . x . x . x . ..., we again operate with δx by striking out an x in all possible ways, and one of the terms so reached is x . x . x . x . x . .... Fixing upon this term, and again operating and continuing the process, we finally arrive at one solution of the problem, which (taking say n = 4) may be said to be in correspondence with the operator diagram—
the number in each row of cempartments denoting an operation of δx. Hence the permutation problem is equivalent to that of placing n units in the compartments of a square lattice of order n in such manner that each row and each column contains a single unit. Observe that the method not only enumerates, but also gives a process by which each solution is actually formed. The same problem is that of placing n rooks upon a chess-board of n² compartments, so that no rook can be captured by any other rook.
Regarding these elementary remarks as introductory, we proceed to give some typical examples of the method. Take a lattice of m columns and n rows, and consider the problem of placing units in the compartments in such wise that the sth column shall contain λs units (s = 1, 2, 3, ... m), and the tth row p1 units (t = l, 2, 3, ... n).
Writing
1 + a1x + a2x² + ... + ... = (1 + a1x) (1 + a2x)(1 + a3x) ...