Proceedings of the London Mathematical Society Volume 19 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1889 Excerpt: ...+ &, 3 3 2 3 Hence, finally, transforming as before to symmetric functions of the roots of the equation xB-M, '-1+Mn, -... = 0. Hr, = (1)' = 1', 2 v ' 2 3 +K 2 + 3' The symmetric function can be thus expressed in terms of ii hi K These results should be compared with those obtained in section 9 for the case of distribution into parcels. It will be noticed that is derived from by simply writing fcr in place of fc-. Section 14. Restricted distributions into Parcels. The distributions considered in the foregoing sections were not subject to any restriction. There was no limit to the number of similar objects that it was permissible to distribute either into a single parcel or into a set of similar parcels. This freedom from restriction led naturally to the invariable appearance of the symmetric functions, which express the sums of the homogeneous products of the quantities, in the distribution functions. In order to find the distribution function of n objects in n parcels, one object in each parcel, subject to the restriction that no two objects of the same kind are to appear in parcels of the same kind, we have merely to employ the elementary symmetric functions i ji -ii instead of the homogeneous product sums 11 h .... The product aPt o4l an... is necessarily the distribution function of n objects into parcels (Pi 1iri )- where?i + g, +r, +... = n, subject to the restriction that no two similar objects are to appear in similar parcels. Thus, since 0,0, = ( ') (I1) = (2'1) +3 (21-) +10 (l8), we discover that, subject to the restriction, objects (21s) can be distributed into parcels (32) in three different ways. These three ways are apparent in the scheme: --We wish now to impose the restriction that not more than t similar objects are ...