(p,q) shuffle

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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 τ 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.

This article incorporates material from (p,q) shuffle on PlanetMath, which is licensed under the GFDL.

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