154
COMBINATORIAL m
veniently written (l ); this is the theory of the compositions of unipartite and multipartite numbers. Case III. is that in which the parcels are all similar, being defined by the partition (m); this is the theory of the partitions of unipartite and multipartite numbers. Previous to discussing these in detail, it is necessary to describe the method of symmetric functions which will be largely utilized. Let a, /?, y,... be the roots of the equation 71-1 71-2 „ xn — oq# + a2x — ... =0. The Distri- The symmetric function '2aP(3qyrwhere button p+q+ ... =n iSj in the partition notation, Function. written (par Let A. . , . denote v/ i ...). 7 _ (mr • • • ), (Pl5in • • •) the number of ways of distributing the n objects defined by the partition (pgr...) into the m parcels defined by the partition (2>1g'1r1...). The expression . . (pqr...), w where the numberspv qv rx... are fixed and assumed to be in descending order of magnitude, the summation being for every partition (pqr ...) of the number n, is defined to be the distribution function of the objects defined by (pqr...) into the parcels defined by (p^ry..). It gives a complete enumeration of n objects of whatever species into parcels of the given species. 1. One-to-One Distribution. Parcels m in number [i.e. m = n).— . Let h, be the homogeneous product-sum of degree s of ase ' the quantities a, y, ... so that (1-ax. l-/3a3. l-7a3. ...)-1 = l+h1x + h^ci + hs^+... A1 = Sa2= (l) 7i2 = 2a3+ 2a/3= (2) + (l2) 2 h3 = 2a + 2a /3 + 2a/37 = (3) + (21) + (l3). Form the product hp-Jig^hri... Any term in hVl may be regarded as derived from pj objects distributed intopj similar parcels, one object in each parcel, since the order of occurrence of the letters a, /3, 7, ... in any term is immaterial. Moreover, every selection of q)l letters from the letters in aP^yr ... will occur in some term of every further selection of qx letters will occur in some term of hq^ and so on. r Therefore in the product hp-Ju^hi^ ... the term aP^y ..., and therefore also the symmetric function (pgr ...) will occur as many times as it is possible to distribute objects defined by (pqr ...) into parcels defined by (pigpfi ...) one object in each parcel. Hence SA (pgr...), (Mm...) - (P2r ••• )=hJ?1hq1hr1 ... This theorem is of algebraic importance ; for consider the simple particular case of the distribution of objects (43) into parcels (52), and represent objects and parcels by small and capital letters respectively. One distribution is shown by the scheme A AAAABB a aaab bb wherein an object denoted by a small letter is placed in a parcel denoted by the capital letter immediately above it. We may interchange small and capital letters and derive from it a distribution of objects (52) into parcels (43) ; viz. :— A A A ABBB, aaa a a b b The process is clearly of general application, and establishes a oneto-one correspondence between the distributions of objects (pqr ...) into parcels (p^j...) and the distributions of objects (p^m ...) into parcels (pqr ...). It is in fact, in Case L, an intuitive observation that we may either consider an object placed in or attached to a parcel, or a parcel placed in or attached to an object. Analytically we have Theorem.—“The coefficient of symmetric function (pqr...) in the development of the product hp1hqlhri... is equal to the coefficient of symmetric function (pigm ...) in the development of the product hphqhr ... .” The problem of Case I. may be considered when the distributions are subject to various restrictions. If the restriction be to the effect that an aggregate of similar parcels is not to contain more than one object of a kind, we haveaclearly to deal with the elementary symmetric functions eq, a2> s> ••• or (^)> (12)> (I8)> ••• in lieu of the quantities h2, h3, ... The distribution function has then the value anaq ary., or (l*1) (1?1) (l"1).... and by inter-
ANALYSIS
change of object and parcel we arrive at the well-known theorem of symmetry in symmetric functions, which states that the coefficient of symmetric function (pqr ...) in the development of the product ap1aq1ari... in a series of monomial symmetric functions, is equal to the coefficient of the function (p^iq ...) in the similar development of the product apaqar.... The general result of Case I. may be further analysed with important consequences. Write X1 = (l)x1, 2 X2=(2)x2 + (1 )x?, X3 = (3 )x3 + (21 )x2x1 + (l3)xi and generally Xs = X(/xv ... )xKx^xu ... the summation being in regard to every partition of s. Consider the result of the multiplication— XPlXgiXn...=2PxVx;3... To determine the nature of the symmetric function P a few definitions are necessary. Definition I.—Of a number n take any partition (A^Ag ... As) and separate it into component partitions thus :— (AiA2)(A3A4A5)(A6) ... in any manner. This may be termed a separation of the partition, the numbers occurring in the separation being identical with those which occur in the partition. In the theory of symmetric functions the separation denotes the product of symmetric functions— 2aAl/3A22aX3/3AV52aX6 ••• The portions (A^), (A3A4A5), (A6), ...are termed separates, and if Ai + A2 =pi, A3 + A4 + A5 = g1, A6 = r1, ... be in descending order of magnitude, the usual arrangement, the separation is said to have a species denoted by the partition (Piq-ir1...) of the number n. Definition II.—If in any distribution of n objects into n parcels (one object in each parcel), we write down a number |, whenever we observe £ similar objects in similar parcels we will obtain a succession of numbers ifi, £2, £3, .. , where (ifi&fs •••) some partition of n. The distribution is then said to have a specification denoted by the partition (f^fs •••)• Now it is clear that P consists of an aggregate of terms, each of which, to a numerical factor pres, is a separation of the partition (s^3...) of species (pqrx ...). Further, P is the distribution function of objects into parcels denoted by (p^r^...), subject to the restriction that the distributions have each of them the specification denoted by the partition (s^s^s^3...). Employing a more general notation we may write X^x^x-s _ _ = S s s ... Pi V‘2. PS 1 2 3 and then P is the distribution function of objects into parcels (p^P^Pg3"-), the distributions being such as have the specification (sps^X3...). Multiplying out P so as to exhibit it as a sum of monomials, we get a result— 3 Cs0-14 X„s(To2 X(T3 x^x^x" S3 ... Pi P2 P3 indicating that for distributions of specification (Yris^V’3..,) 3 there are 0 ways of distributing n objects denoted by (A^A^A^ ...) amongst n parcels denoted by (p^V^2P33---)> one object in each parcel. Now observe that as before we may interchange parcel and object, and that this operation leaves the specification of the distributions unchanged. Hence the number of distributions must be the same, and if x«ikes s L.. + ... X^X^X"3 2 3 Pi P2 P3 then also x x A1 K-=-+«o»r"->K<;-+-
This extensive theorem of algebraic reciprocity includes many known theorems of symmetry in the theory of Symmetric Functions. The whole of the theory has been extended to include symmetric functions symbolized by partitions which contain as well zero and negative parts. 2. The Compositions of Multipartite Numbers. Parcels denoted by (l»n).—There are here no similarities between the parcels.
