(p,q) shuffle

From Wikipedia, the free encyclopedia

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.

(p,q) shuffle

From Wikipedia, the free encyclopedia

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