MacMahon Percy Alexander 1854-1929's An introduction to combinatory analysis PDF

By MacMahon Percy Alexander 1854-1929

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The case k=\ is interesting because the homogeneous productsums become the elementary functions (1), (1-), (I'O, and we ... , establish that the coefficient of the function (pi2>2 ••ps) in the development of is the same as that of (qiq^ • qt) in the development of This particular case of symmetric function symmetry has been known Meyer Hirsch and That here given, based upon the theory of distribution, is the simplest and most suggestive. Since the specification of one of these distributions cannot involve any number greater than unity, we see that every distribution must have the same specification, viz.

Ence there is boxes of specification ii^to being all different, one object being number of distributions of objects of into boxes of specification (t) such that the qt objects respectively. ,... qt) t For example, compare these distributions where {]hP2 hQ objects will only qi, q^, ••- qt objects. ^, In con- any pair of the Also in the second distribution change of a pair of the sets of (^1^2 q^, q^, in the first distribution interchange the qr similar boxes Ar. Bi, numbers If the Bt respectively.

For a given weight the rows Thus such a hK If 1 1 12 (2P) is called 1 1 2 3 4 ^2' 1 2 3 4 6 1 3 4 7 12 ^/ 1 4 6 12 24 we look again at the read the same is (1^) 1 hK- distribution AAA BB C a a y a a will table for the weight four (4) (31) (2^) /^4 43. This hp^. , y ; so that 39 we may regard the distribution as being in fact a pairing of objects of two different sets of objects, one object being taken from each set to form a pair. Observe that one set of objects involves no objects which appear in the other set.

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

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