By MacMahon Percy Alexander 1854-1929

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**Additional info for An introduction to combinatory analysis**

**Example text**

Bi, B2, each box. in qt objects placed in single boxes - (321), ARE EQUAL Again, 51. qt objects can be permuted in all possible — ways so as to produce new distributions the number of ways depending upon the similarities that may exist between the t sets of objects. Also in the second distribution, since the boxes are all different, the sets of objects can be may permuted exactly as in the first distribution, {piPo . p^ into boxes of specification {q-iq-i • • • qt), the numbers qi,q2,---qt being identical, one object being placed in each box, number of and we say that the number of distributions of objects of specification of distributions of objects of specification (pip^ specification qii q2i ••• qt (1*) such that the objects respectively.

X^^ + . and the distributions are B C ft a ft ft ft A A A A A 3 in number and each 57. The symbol we operate upon a a a ^ a a a ft a a y3 /8 jS a a a of specification (321"). D,,^, we obtain a linear, function oi x products which gives a complete speci- fication account of the distributions of the objects into the boxes. s = if we put Xo~ and recall the way in which the symbol operates upon a product through the compositions of calculation 1. ,x{' + x^') X^ = Di {Ixix-r + ^x^x-i* + 3«'i*)Xi ] { { \ 2>,„ The suffix.

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.