Page:1902 Encyclopædia Britannica - Volume 27 - CHI-ELD.pdf/184

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COMBINATORIAL ANALYSIS Suppose that we wish to find the generating function for the the collection, termed the parts of the partition, in descending order of magnitude, and to indicate repetitions of the same part enumeration of those permutations of the letters in by the use of exponents. Thus (32111), a partition of 8, is written which are such that no letter a^is in a position originally occupied (3213). Euler’s pioneering work in the subject rests on the observaby an x, for all values of s. This is a generalization of the tion that the algebraic multiplication “Probleme des rencontres” or of “derangements.” We have XaXxh XX^X ... =£Ca+i+c+... merely to put == =z is equivalent to the arithmetical addition of the exponents CCn — 0^22 6^33 * - • ~ &nn — a, b, c, .... He showed that the number of ways of composing n and the remaining elements equal to unity. The generating with p integers drawn from the series a, b, c, ..., repeated or product is not, is equal to the coefficient of p)xn in the ascending expansion of the fraction {x2 + X3 + ...+ + aj3 + ... + Xnf2...(»! + JBa +... + xn-{fn, 1 and to obtain the condensed form we have to evaluate the co-axial l-£c“. l-fx4. l-fxc. ...’ minors of the invertebrate determinant— which he termed the generating function of the partitions in 0 1 1 ... 1 question. 1 0 1 ... 1 If the partitions are to be composed of p, or fewer parts, it is 1 1 0 ... 1 merely necessary to multiply this fraction by Similarly, if i 1 1 ... 0 the parts are to be unrepeated, the generating function is the The minors of the 1st, 2nd, 3rd ... fith orders have respectively the algebraic product values (l + fxa)(l + fx4)(l + i'xc)..., 0 if each part may occur at most twice, -1 +2 (1 + + CVS)(1 + (tc* + (Vc)..., and generally if each part may occur at most k-1 times it is (-)«-!(„-1), k ka 1 — £ x 1 — 'fixkbb 1 — 0 therefore the generating function is 1 ("x“ ’ 1 £x ' 1 - (ix 1 It is thus easy to form generating functions for the partitions of 1- 2'^x1x^c3 - ... - s2x1x2...a^s+1 - ... -{n- l)x1Xo...2:n numbers into parts subject to various restrictions. If there be no or writing restriction in regard to the numbers of the parts, the generating function is {x-xj){x-xj)...{x-xn) = xn-a1xn~1 + a% 2- ... 1 this is 1—xh. —xb. 1—x®. ... 1 and the problems of finding the partitions of a number n, and of 1 — 0^2 — ^0^3 Sct/^ ... — 1 'jCl/n their number, are the same as those of solving and Again, consider the general problem of “derangements.” We determining the solutions of the indeterminate equation in have to find the number of permutations such that exactly m of enumerating the letters are in places they originally occupied. We have the positive integers ax+by + cz+ ... —n. particular redundant product Euler considered also the question of enumerating the solutions of the indeterminate simultaneous equation in positive integers (axl + x.2+ ...+xnfx1 + axz+ ... + xn)^..-(xx+x2+ ... + axn'f’n, ax + by + cz+...=n in which the sought number is the coefficient of anx^x^...xr^. a'x + &'^ + c'z +... = »' The true generating function is derived from the determinant al'x+b"y + c"z +... = n" a 1 1 1 . . . fall... which was called by him and those of his time the “ Problem of 1 1 a 1 . . . the Virgins.” The enumeration is given by the coefficient of 111a... xnyn'zn"... in the expansion of the fraction 1 and has the form (1 -x“2/V...)(l -xh'yh'fi ...)(l-xr"yb"zf"...)... which enumerates the partitions' of the multipartite number 1 -112*1+(a - l)(a+l)2xi*2 -..+(- )"(tl - L” _ V+™ nn'n"... into the parts It is clear that a large class of problems in permutations abc..., a'b’c’..., a"b"c" can be solved in a similar manner, viz., by giving special Sylvester has determined an analytical expression for the covalues to the elements of the determinant of the matrix. efficient of x” in the expansion of The redundant product leads uniquely to the real generat1 ing function, but the latter has generally more than one (1 -xa)(l —Xs) ... (1 — Xs)’ representation as a redundant product, in the cases in To explain this we have two lemmas :— which it is representable at all. For the existence of a Lemma 1.—The coefficient of i, i.e., after Cauchy, the residue redundant form, the coefficients of xv x2, ... ... in the denominator of the real generating function must satisfy in the ascending expansion of (1 - ex)~l is -1. For when i is unity, and 2n -n2 + n- 2 conditions, and assuming this to be the it is obviously the case, (1 - e*)-1-1 = (1 - e*)-4 + ex(l - e*)-4-1 case, a redundant form can be constructed which involves n — 1 undetermined quantities. We are thus able to pass from any particular redundant generating function to one equivalent to it, but involving n-1 undetermined Here the residue of ^ (1 - e1)-4 is zero, and therefore the residue quantities. Assuming these quantities at pleasure we of (1 - e*)-4, is unchanged when i is increased by unity, and is, obtain a number of different algebraic products, each of therefore, always - 1 for all values of i. which may have its own meaning in arithmetic, and thus Lemma 2.—The constant term in any proper algebraical fraction the number of arithmetical correspondences obtainable is developed in ascending powers of its variable is the same as the with changed sign, of the sum of the fractions obtained subject to no finite limit (cf. MacMahon, loc. cit., pp. 125 residua, by substituting in the given fraction, in lieu of the variable, its et seq.y exponential multiplied in succession by each of its values (zero 3. The Theory of Partitions. Parcels defined by (m).—When an excepted, if there be such), which makes the given fraction infinite. ordinary unipartite number n is brokenof up other For write the proper algebraical fraction Case III. numbers, and the order of occurrence theinto numbers c 7a Fx=22 (a x,* + aA’ is immaterial, the collection of numbers is termed a partition of M-x)A the number n. It is usual to arrange the numbers comprised in