Suppose that we wish to find the generating function for the enumeration of those permutations of the letters in xξ11 xξ22 ... xξnn 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 ... (x1 + x2 + ... + xn-1)ξn,

and to obtain the condensed form we have to evaluate the co-axial minors of the invertebrate determinant—

011...1
101...1
110...1
.......
111...0

The minors of the 1st, 2nd, 3rd ... nth orders have respectively the values

0
-1
+2
.
.
.
(-)n-1 (n - 1),

therefore the generating function is

1;
1 - Σ x1x2 -2Σ x1x2x3 - ... -sΣ x1x2 ... xs+1 - ... -(n - 1) x1x2 ... xn