relation

(X1, X2, X3, ... Xn) = (a11 a12 ... a1n)(x1, x2, ... xn)
|a21 a22 ... a2n|
| . . ... . |
| . . ... . |
|an1 an2 ... ann|

that portion of the algebraic fraction,

1
---------------------------------,
(1 - s1X1)(1 - s2X2)...(1 - snXn)

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

1
--------------------------------------------------------
|(1 - a11s1x1)(1 - a22s2x2)(1 - a33s3x3)(1 - ann.sn.xn)|

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

1 - [Sigma]|a11|s1x1 + [Sigma]|a11a22|s1s2x1x2 - ... + (-)^n|a11a22a33...ann|s1s2 ... sn.x1x2...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

x1^[xi]1 x2^[xi]2 ... xn^[xi]n

in the product

(a11x1 + a12x2 + ... + a1n.xn )^[xi]^1 (a21x1 + a22x2 + ... +
+ a2n.xn)^[xi]^2...(an1x1 + an2x2 + ... + ann.xn)^[xi]n

is equal to the coefficient of the same term in the expansion
ascending-wise of the fraction

1
--------------------------------------------------------------------------.
1 - [Sigma]|a11|x1 + [Sigma]|a11a22|x1x2 - ... + (-)^n|a11a22...|x1x2...xn

If the elements of the determinant be all of them equal to unity, we
obtain the functions which enumerate the unrestricted permutations of
the letters in

x1^[xi]1 x2^[xi]2 ... xn^[xi]n,

viz. (x1 + x2 + ... - xn)^{[xi]1 + [xi]2 + ... + [xi]n}

1
and ------------------------.
1 - (x1 + x2 + ... + xn)

Suppose that we wish to find the generating function for the enumeration
of those permutations of the letters in x1^[xi]1 x2^[xi]2...x3^[xi]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 "Probleme 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 + x3 + ... + xn)^[xi]1 (x1 + x3 + ... + xn)^[xi]2 ...
(x1 + x2 + ..