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By Clayton R. Newell

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Extra info for Central Pacific: The United States Army Campaigns of World War 2 (CMH pub)

Example text

2 The same reasoning ... iW2 applies ... s) be distributed into boxes of specification (p+-'+*+-) in such wise that i unspecified boxes receive {mim2 •• w^^)> J other unspecified boxes objects ^ other unspecified boxes objects of specifica- objects of specification of specification tion (ojOo ... , (9i,), Wm)> ••• number •,•/. o,y.... y) = 2a='y8^y"- + 22a«/32y8 + GSa^'ySySc + 3Xa'l3'y^8 + eSa^^^ySc + lOSa^ySySc^, otherwise written (2P) 28. ) + 6 (2-P) + 10 (2r). important to be able to obtain readily the numerical values of such coefficients, we will subject this particular development to examination with the object of deducing general laws in the algebra of symmetric functions.

Observe (2P) (P) the highest power of a thai presents are the largest parts in the factors respectively OPENING OF THE THEORY OF DISTRIBUTIONS 24 and 2 + 1 = 3. It follows at once that the coefficient of a^ is found by simply obliterating the first or largest part We in the product in each factor. Thus from the original identity (P) = (32') + 2 (321") + 6 (31") + other terms which involve no part, thus arrive at the coefficient (1^)^ (21-) we in the partitions, as large as 3, (P)- = (2^) + From derive, at sight, the As an example, in distribution.

The enumeration coefficient of the function hq^ in a/iao^'-^ ... a/« of the distributions (piP2 Ps) in the a series of monomial symmetric functions. DISTRIBUTION WHEN OBJECTS AND BOXES EQUAL IN NUMBER 37 "The number of ways of distributing n objects of specification p^ into n boxes of specification {q-^q^ ... qi), one object into (i>ii>2 each box, in the is equal to the coefficient of symmetric function development of the product As an example, distribution if (io,B h^Ji^^ •••JOs) = ... X A2 -A^ Az a a a a (3 y corresponding to the terms of distributions is and the complete a^, a(3, y in ^3, L, hi respectively.

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