PlanetMath: $(p,q)$ shuffle

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shuffle

(Definition)

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

$\displaystyle \tau(1)<$	$\displaystyle \cdots$	$\displaystyle < \tau(p),$
$\displaystyle \tau(p+1)<$	$\displaystyle \cdots$	$\displaystyle < \tau(p+q).$

The set of all shuffles is denoted by .

It is clear that $S(p,q)\subset S(p+q)$ . Since a shuffle is completely determined by how the first elements are mapped, the cardinality of is ${p+q \choose p}$ . The wedge product of a -form and a -form can be defined as a sum over shuffles.

shuffle" is owned by mathcam. [ owner history (1) ]

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shuffle

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Cross-references: sum, wedge product, cardinality, clear, numbers, permutations, natural numbers, positive
There are 3 references to this entry.

This is version 4 of $(p,q)$

shuffle, born on 2003-04-10, modified 2004-02-21.
Object id is 4176, canonical name is PqShuffle.
Accessed 3386 times total.

Classification:

AMS MSC:	20B99 (Group theory and generalizations :: Permutation groups :: Miscellaneous)
	05A05 (Combinatorics :: Enumerative combinatorics :: Combinatorial choice problems )

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