(p,q) shuffle
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Let <math>p</math> and <math>q</math> be positive natural numbers. Further, let <math>S(k)</math> be the set of permutations of the numbers <math>\{1,\ldots, k\}</math>. A permutation <math>\tau</math> in <math>S(p+q)</math> is a (p,q)shuffle if
- <math> \tau(1) < \cdots < \tau(p) \,</math>,
- <math> \tau(p+1) < \cdots < \tau(p+q) \,</math>.
The set of all <math>(p,q) </math> shuffles is denoted by <math>S(p,q).</math>
It is clear that
- <math>S(p,q)\subset S(p+q).</math>
Since a <math>(p,q) </math> shuffle is completely determined by how the <math>p</math> first elements are mapped, the cardinality of <math>S(p,q)</math> is
- <math>{p+q \choose p}.</math>
The wedge product of a <math>p</math>-form and a <math>q</math>-form can be defined as a sum over <math>(p,q) </math> shuffles.
This article incorporates material from (p,q) shuffle on PlanetMath, which is licensed under the GFDL.
