Percy Alexander MacMahon

“ The distributions to be considered are such that any number of objects may be in any one class subject to the restriction that no class is empty. Two cases arise. If the order of the objects in a particular class is immaterial, the class is termed a parcel; if the order is material, the class is termed a group. The distribution into parcels is alone considered here, and the main problem is the enumeration of the distributions of objects defined by the partition (pqr . . . ) of the number n into parcels defined by the partition (p1q1r1 . . . ) of the number m. ”

“ The partition method of treating symmetrical algebra is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important theorem of expressibility is, directly we exclude unity from the partitions, a theorem concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies. ”

“ The redundant product leads uniquely to the real generating function, but the latter has generally more than one representation as a redundant product, in the cases in which it is representable at all. ”

“ 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. ”

“ Symmetric Functions of Several Systems of Quantities.—It will suffice to consider two systems of quantities as the corresponding theory for three or more systems is obtainable by an obvious enlargement of the nomenclature and notation.
Taking the systems of quantities to be
we start with the fundamental relation
As shown by L. Schläfli [2] this equation may be directly formed and exhibited as the resultant of two given equations, and an arbitrary linear non-homogeneous equation in two variables. ”

“ They are the chief instruments of research, and have themselves much benefited by being so employed. When a homogeneous polynomial is transformed by general linear substitutions as hereafter explained, and is then expressed in the original form with new coefficients affecting the new variables, certain functions of the new coefficients and variables are numerical multiples of the same functions of the original coefficients and variables. ”

“ 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 quantities are called the elements of the determinant; the term is called a member of the determinant, and there are evidently members corresponding to the permutations of the numbers The determinant is usually written
the square array being termed the matrix of the determinant. A matrix has in many parts of mathematics a signification apart from its evaluation as a determinant. A theory of matrices has been constructed by Cayley in connexion particularly with the theory of linear transformation. The matrix consists of rows and columns. ”

“ A binary form which has a square factor has its discriminant equal to zero. This can be seen at once because the factor in question being once repeated in both differentials, the resultant of the latter must vanish.
Similarly, if a form in k variables be expressible as a quadratic function of k – 1, linear functions X1, X2, ... Xk – 1, the coefficients being any polynomials, it is clear that the k differentials have, in common, the system of roots derived from X1=X2=...=Xk – 1=0, and have in consequence a vanishing resultant. ”

“ To illustrate an important dissection of the graph we will consider those graphs which read the same by columns as by lines; these are called self-conjugate. Such a graph may be obviously dissected into a square, containing say θ2 nodes, and into two graphs, one lateral and one subjacent, the latter being the conjugate of the former. The former graph is limited to contain not more than θ parts, but is subject to no other condition. ”

“ 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 . . .) ”

“ A skew symmetric determinant has and for all values of and . Such a determinant when of uneven degree vanishes, for if we multiply each row by we multiply the determinant by , and the effect of this is otherwise merely to transpose the determinant so that it reads by rows as it formerly did by columns, an operation which we know leaves the determinant unaltered. Hence or . When a skew symmetric determinant is of even degree it is a perfect square. This theorem is due to Cayley, and reference may be made to Salmon’s Higher Algebra, 4th ed. Art. ”

“ 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.) ”

“ Expanding the right-hand side by the exponential theorem, and then expressing the symmetric functions of which arise, in terms of we obtain by comparison with the middle series the symbolical representation of all symmetric functions in brackets appertaining to the quantities To obtain particular theorems the quantities are auxiliaries which are at our entire disposal. ”

“ 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 − ...) . ”

“ Now , wherein is not to be changed, but the second suffixes in the product assume all permutations, the number of transpositions necessary determining the sign to be affixed to the member.
Hence , where the cofactor of is clearly the determinant obtained by erasing the first row and the first column.
Hence
Similarly , the cofactor of , is shown to be the product of and the determinant obtained by erasing from the i th row and k th column. No member of a determinant can involve more than one element from the first row. ”

“ 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 . ”

“ Operation of upon a Product of Symmetric Functions.—Suppose to be a product of symmetric functions If in the identity we introduce a new root we change into and we obtain
and now expanding and equating coefficients of like powers of
the summation in a term covering every distribution of the operators of the type presenting itself in the term.
Writing these results
we may write in general
the summation being for every partition of and being
Ex. gr. To operate with upon we have
and hence
Application to Symmetric Function Multiplication.—An example will explain this. ”

“ Putting and we find a particular law of reciprocity given by Cayley and Betti,
and another by putting , for then becomes , and we have
Theorem of Expressibility.—“If a symmetric function be symboilized by and be any partitions of respectively, the function is expressible by means of functions symbolized by separation of
For, writing as before,
is a linear function of separations of of specification and if is a linear function of separations of of specification Suppose the separations of to involve different specifications and form the identities
where is one of the specifications. ”

“ 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. ”

“ Thus to obtain Stroh’s theory of seminvariants put
we then obtain the expression of non-unitary symmetric functions of the quantities as functions of differences of the symbols
Ex. gr. with must be a term in
and since we must have
as is well known.
Again, if be the roots of and leading to
and
and we see further that vanishes identically unless . If be infinite and
we have the symbolic identity
and
Instead of the above symbols we may use equivalent differential operators. Thus let
and let be equivalent quantities. ”

“ 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. ”

“ American Journal of Mathematics, Baltimore, Md. 1888–1890; “Memoir on Symmetric Functions of Roots of Systems of Equations,” Phil. Trans. 1890.
III. The Theory of Binary Forms
A binary form of order n is a homogeneous polynomial of the nth degree in two variables. It may be written in the form
axn
1 + bxn−1
1x2 + cn−2
1x2
2 + ...;
or in the form
axn
1 + (n
1) bxn−1
1x2 + (n
2) cn−2
1x2
2 + ...;
which Cayley denotes by
(a, b, c, ...) (x1, x2) n
(n
1) , (n
2) ... being a notation for the successive binomial coefficients n, 1/2n (n − 1) , .... ”

“ At various times he was president of the Cambridge Philosophical Society, of the London Mathematical Society and of the Royal Astronomical Society. He was elected a fellow of the Royal Society in 1852, and received from that body a Royal medal in 1859 and the Copley medal in 1882. He also received the De Morgan medal from the London Mathematical Society, and the Huygens medal from Leiden. ”

“ CAYLEY, ARTHUR (1821–1895) , English mathematician, was born at Richmond, in Surrey, on the 16th of August 1821, the second son of Henry Cayley, a Russian merchant, and Maria Antonia Doughty. His father, Henry Cayley, retired from business in 1829 and settled in Blackheath, where Arthur was sent to a private school kept by the Rev. ”

“ Any function of differences of being formed, the expansion being carried out, an operand or or being taken and being subsequently put equal to , a non-unitary symmetric function will be produced.
Ex. gr. ”

“ Since
dpq = ( − ) p+q−1 (p + q − 1) !/p!q!d/dspq
the solutions of the partial differential equation dpq = 0 are the single bipart forms, omitting spq, and we have seen that the solutions of Dpq = 0 are those monomial functions in which the part pq is absent.
One more relation is easily obtained, viz.
d/dapq = dpq − h10dp+1,q − h01dp,q+1 + ... + ( − ) r+shrsdp+r,q+s + ... .
References for Symmetric Functions.—Albert Girard, Invention nouvelle en l’algèbre (Amsterdam, 1629) ; Thomas Waring, Meditationes Algebraicae (London, 1782) ; Lagrange, Mém. de l’acad. de Berlin (1768) ”