in the product
(a11x1 + a12x2 + ... + a1nxn)ξ1 (a21x1 + a22x2 + ... + a2nxn)ξ2 ... (an1x1 + an2x2 + ... + annxn)ξn
is equal to the coefficient of the same term in the expansion ascending-wise of the fraction
| 1 | . |
| 1 - Σ |a11|x1 + Σ |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
xξ11 xξ22 ... xξnn,
viz.
(x1 + x2 + ... - xn)ξ1+ξ2+ ... +ξn
and
| 1 | . |
| 1 - (x1 + x2 + ... + xn) |