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# $(p,q)$ shuffle

###### Definition.

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

$\displaystyle\tau(1)<$ | $\displaystyle\cdots$ | $\displaystyle<\tau(p),$ | ||

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

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.

Related:

ShuffleOfLanguages

Synonym:

shuffle

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

20B99*no label found*05A05

*no label found*

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new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

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new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

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new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier