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By MacMahon Percy Alexander 1854-1929

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Mif if the partitions im^ mo have merely to develop the product and seek the the function We nit) (tii »2 m^, ^m) be identical. now coefficient of {p-ip-2---p^- abolish the restriction and substitute another, viz. that the boxes are to receive, one of them w^ objects and the other W2 objects. 2 '^' specification objects and iv^ must have the same If the specification of the objects. and the iv^ objects consistent with the condition that the w specification as objects which are to be unknown the Wi and Wo objects may have specifications denoted by any partitions of w^ and w^ respectively.

If the objects of both sets had been drawn from one set of objects, so that the objects in one set were not distinct from the objects in the other set, the distribution theory considered here For example, not be valid. boxes A, B, objects but if C we A, B, if we obtain the six pairings of the objects ABC ABC a « y /5 we y /3 ay pair off the identical sets «, five pairings a^y a, )S, a, jB, would y into the y with the C, ABC 13 distribute the objects ajSy a;Sy ABC ABC ABC /S y'^ft y^^ y )S, <^ y; a, ^, y, we obtain only the 40 DISTRIBUTION Or, we may say that WHEN OBJECTS AND BOXES objects of specification (411) have been paired with other objects of specification (321) in such wise that the off specification of the object-pairs is (3111).

I) (1) (1), = 2 (i), finally AAA^(2P)(P) = 2A(1) = Now we may write (21^)(P) so that operating becomes The C since 2. , sides with every other term is AAA" the right-hand side reduced to zero by the operation. calculation above shews that the left-hand side becomes 2 operation. Hence C = 2, (2P)(r') = We number by the and --- + 2(32P)-f .... can in this way calculate the result of the product of any of monomial functions and thus evaluate the number which enumerates a well-defined distribution of objects into boxes.

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An introduction to combinatory analysis by MacMahon Percy Alexander 1854-1929

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