F(22) = 2 (2² + 2 + 2 + 2 + 2 + 2°) = 26.

We may regard the fraction

½ · 1 ,
{1 - t1 (2α1 + α2 + ... + αs)} {1 - t2 (2α1 + 2α2 + ... + αs)} ... {1 - ts (2α1 + 2α2 + ... + 2αs)}

as a redundant generating function, the enumeration of the compositions being given by the coefficient of

(t1α1)p1 (t2α2)p2 ... (tsαs)ps.

The transformation of the pure generating function into a factorized redundant form supplies the key to the solution of a large number of questions in the theory of ordinary permutations, as will be seen later.

[The transformation of the last section involves The theory of permutations. a comprehensive theory of Permutations, which it is convenient to discuss shortly here.

If X1, X2, X3, ... Xn be linear functions given by the matricular relation

(X1, X2, X3, ... Xn) =(a11a12...a1n)(x1, x2, ... xn)
a21a22...a2n
......
......
an1an2...ann

that portion of the algebraic fraction,