An Introduction to Combinatory Analysis by MacMahon Percy Alexander 1854-1929

By MacMahon Percy Alexander 1854-1929

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Butions are there Suppose that ? q^ of the boxes are precisely similar, so that they have the specification {q^. Whatever may be the specification of the q^ objects that are placed in them it is certain that they have only one no permutation of because the boxes being identical distribution, Denote these boxes each by u4i. The specification of the q^ objects must be one of the partitions which occur in h^^ when expressed in terms of monomial functions. As the objects alters the distribution. one distribution we Also in hq^.

In kq^kq, . k,i^ is ^'ps- kp^ kp^ ... kp^. Moreover, since JJpK^^= also Kjfi-p, the evaluation of the coefficients can be carried out. The specification of the distribution is clearly not altered by the interchange of Capital and Greek letters and we are led to an extended theory of symmetry in the Algebra of Symmetric Functions. 47. 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.

Bi,B2,.. respectively. This implies that the boxes being in a definite order the i+j combinations is to say that tions is of objects are only allowed iljl permutations; that no exchange of combinations of objects of different specifica- allowed to take place. If such exchange be permitted permutations between the combinations of objects function that by now be its (/ +j)l may take place. ps) be distributed into boxes of specification (P"^-') in such wise that i of the boxes (unspecified) receive OPENING OF THE THEORY OF DISTRIBUTIONS and the remaining boxes objects of objects of specification (1") number fication (1), the the function {piP^.

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