Page:EB1911 - Volume 06.djvu/776

Rh

denoted by the partition $$\big(s_1^{\sigma_1}s_2^{\sigma_2}s_3^{\sigma_3}...\big).$$ Employing a more general notation we may write

and then P is the distribution function of objects into parcels (p p p ...), the distributions being such as to have the specification $$\big(s_1^{\sigma_1}s_2^{\sigma_2}s_3^{\sigma_3}...\big).$$. Multiplying out P so as to exhibit it as a sum of monomials, we get a result—

indicating that for distributions of specification $$\big(s_1^{\sigma_1}s_2^{\sigma_2}s_3^{\sigma_3}...\big).$$. there are ways of distributing n objects denoted by (   ... ) amongst n parcels denoted by (p11 p22 p33 ...), 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 distribution unchanged. Hence the number of distributions must be the same, and if

X X X ... ... = ... + (   ...) x x x ... + ...

then also

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'' (Im).—There are here no similarities between the parcels.

Let (1 2 3) be a partition of m.

(p p p  ...) a partition of n.

Of the whole number of distributions of the n objects, there will be a certain number such that n1 parcels each contain p1 objects, and in general s parcels each contain ps objects, where s = 1, 2, 3, ... Consider the product h h  h  ... which can be permuted in $m!⁄_{1}!_{2}!_{3}! ...$ ways. For each of these ways h h  h  ... will be a distribution function for distributions of the specified type. Hence, regarding all the permutations, the distribution function is

and regarding, as well, all the partitions of n into exactly m parts, the desired distribution function is

that is, it is the coefficient of xn in (h1x + h2x² + h3x³ + ... )m. The value of A(p1 1 p2 2 p3 3 ...), (1m) is the coefficient of (p1 1 p2 2 p3 3 ...)xn in the development of the above expression, and is easily shown to have the value

Observe that when p1 = p2 = p3 = ... = 1 = 2 = 3 ... = 1 this expression reduces to the mth divided differences of 0n. The expression gives the compositions of the multipartite number p11 p22 p33 ... into m parts. Summing the distribution function from m = 1 to m = &infin; and putting x = 1, as we may without detriment, we find that the totality of the compositions is given by $h_{1} + h_{2} + h_{3} + ...⁄1 − h_{1} − h_{2} − h_{3} + ...$ which may be given the form $a_{1} − a_{2} + a_{3} − ...⁄1 − 2(a_{1} − a_{2} + a_{3} − ...)$. Adding we bring this to the still more convenient form

Let F (p11 p22 p33 ... ) denote the total number of compositions of the multipartite p11 p22 p33 .... Then $1⁄1 − 2a$ =  +  F(p)p, and thence F(p) = 2p − 1. Again · $1⁄1 − 2( + β − )$ = F(p1p2) p1p2, and expanding the left-hand side we easily find

We have found that the number of compositions of the multipartite p1p2p3 ... ps (p1p2p3 ... ps) or of the single term ...  in the development according to ascending powers of the algebraic fraction

This result can be thrown into another suggestive form, for it can be proved that this portion of the expanded fraction

which is composed entirely of powers of

t11, t22, t33, ... tss

has the expression

and therefore the coefficient of p1 1 p2 2 ... ps s in the latter fraction, when t1, t2, &c., are put equal to unity, is equal to the coefficient of the same term in the product

(21 + 2 + ... + s)p1 (21 + 22 + ... + s)p2 ... (21 + 22 + ... + 2s)ps.

This result gives a direct connexion between the number of compositions and the permutations of the letters in the product ...  . Selecting any permutation, suppose that the letter ar occurs qr times in the last pr + pr+1 + ... + ps places of the permutation; the coefficient in question may be represented by 2q1+q2+ ... +qs, the summation being for every permutation, and since q1 = p1 this may be written

2p1−1 2q1+q2+ ... +qs.

Ex. Gr.—For the bipartite 22, p1 = p2 = 2, and we have the following scheme:—

Hence

We may regard the fraction

as a redundant generating function, the enumeration of the compositions being given by the coefficient of

(t11)p1 (t22)p2 ... (tss)ps.

The transformation of the pure generating function into a factorized redundant form supplies the key to the solution of a large number of questions in the theory of ordinary permutations, as will be seen later.

[The transformation of the last section involves a comprehensive theory of Permutations, which it is convenient to discuss shortly here.

If X1, X2, X3, ... Xn be linear functions given by the matricular relation

that portion of the algebraic fraction,

which is a function of the products s1x1, s2x2, s3x3, ... snxn only is

where the denominator is in a symbolic form and denotes on expansion

1 − |a11|s1x1 + |a11a22|s1s2x1x2 − ... + (–)n |a11a22a33 ... ann| s1s2 ... snx1x2 ... xn,

where |a11|, |a11a22|, ... |a11a22, ... ann| denote the several co-axial minors of the determinant

|a11a22 ... ann|

of the matrix. (For the proof of this theorem see MacMahon, “A certain Class of Generating Functions in the Theory of Numbers,” Phil. Trans. R. S. vol. clxxxv. A, 1894). It follows that the coefficient of

x x x