Definitions

mind p q

(p,q) shuffle

Let p and q be positive natural numbers. Further, let S(k) be the set of permutations of the numbers {1,ldots, k}. A permutation tau in S(p+q) is a (p,q)shuffle if

tau(1) < cdots < tau(p) ,,
tau(p+1) < cdots < tau(p+q) ,.

The set of all (p,q) shuffles is denoted by S(p,q).

It is clear that

S(p,q)subset S(p+q).

Since a (p,q) shuffle is completely determined by how the p first elements are mapped, the cardinality of S(p,q) is

{p+q choose p}.

The wedge product of a p-form and a q-form can be defined as a sum over (p,q) shuffles.

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