Percy Alexander MacMahon

“ Problems of combination were generally undertaken as they became necessary for the advancement of some particular part of mathematical science: it was not recognized that the theory of combinations is in reality a science by itself, well worth studying for its own sake irrespective of applications to other parts of analysis. There was a total absence of orderly development, and until the first third of the 19th century had passed, Euler’s classical paper remained alike the chief result and the only scientific method of combinatorial analysis. ”

“ In the simplest example of the three-dimensional graph we have to solve the system
and a system for the general lattice constructed upon the same principle. The system has been discussed by MacMahon, Phil. Trans. vol. clxxxvii. A, 1896, pp. 619-673, with the conclusion that if the numbers of nodes along the axes of x, y, z be limited not to exceed the numbers m, n, l respectively, then writing for brevity 1 – xs = (s) , the generating function is given by the product of the factors
one factor appearing at each point of the lattice. ”

“ 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 have clearly to deal with the elementary symmetric functions a1, a2, a3, . . . or (1) , (12) , (13) , . . . in lieu of the quantities h1, h2, h3, . . . The distribution function has then the value ap1aq1ar1... or (1p1) (1q1) (1r1) ..., and by interchange 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 ap1aq1ar1 . ”

“ Suppose that we wish to find the generating function for the enumeration of those permutations of the letters in xξ1
1 xξ2
2 ... xξn
n which are such that no letter xs is in a position originally occupied by an x3 for all values of s. This is a generalization of the “Problème des rencontres” or of “derangements.” We have merely to put
a11 = a22 = a33 = ... = ann = 0
and the remaining elements equal to unity. The generating product is
(x2 + x2 + ... + xn) ξ1 (x1 + x3 + ... + xn) ξ2 ... ”

“ The most important problem of combinatorial analysis is connected with the distribution of objects into classes. A number n may be regarded as enumerating n similar objects; it is then said to be unipartite. On the other hand, if the objects be not all similar they cannot be effectively enumerated Fundamental problem. by a single integer; we require a succession of integers. If the objects be p in number of one kind, q of a second kind, r of a third, &c., the enumeration is given by the succession pqr . ”

“ Form the product hp1hq1hr1...
Any term in hp1 may be regarded as derived from p1 objects distributed into p1 similar parcels, one object in each parcel, since the order of occurrence of the letters α, β, γ, . . . in any term is immaterial. Moreover, every selection of p1 letters from the letters in αpβqγr . . . will occur in some term of hp1, every further selection of q1 letters will occur in some term of hq1, and so on. Therefore in the product hp1hq1hr1 . . . the term αpβqγr . . ., and therefore also the symmetric function (pqr . . .) ”

“ His work on Probabilities would naturally lead him to consider questions of this nature. An important work at the time it was published was the De Partitione Numerorum of Leonhard Euler, in which the consideration of the reciprocal of the product (1 − xz) (1 − x2z) (1 − x3z) . . . establishes a fundamental connexion between arithmetic and algebra, arithmetical addition being made to depend upon algebraical multiplication, and a close bond is secured between the theories of discontinuous and continuous quantities. (Cf. Numbers, Partition of.) ”

“ Summing the distribution function from m = 1 to m = ∞ and putting x = 1, as we may without detriment, we find that the totality of the compositions is given by h1 + h2 + h3 + .../ 1 − h1 − h2 − h3 + ... which may be given the form a1 − a2 + a3 − .../1 − 2 (a1 − a2 + a3 − ...) . ”

“ Some of the elementary theorems and various particular problems appear in the works of the earliest algebraists, but the true pioneer of modern researches seems to have been Abraham Demoivre, who first published in Phil. Trans. (1697) the law of the general coefficient in the expansion of the series a + bx + cx2 + dx3 + . . . raised to any power. ”

“ This was a distinct advance, and he was able to obtain expressions for the coefficients in partition series in some of the simplest cases (Trans. Camb. Phil. Soc., 1849) . This paper inspired a valuable paper by Sir John Herschel (Phil. Trans. 1850) , who, by introducing the idea and notation of the circulating function, was able to present results in advance of those of Warburton. The new idea involved a calculus of the imaginary roots of unity. ”