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Math Help - permutations

  1. #1
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    permutations

    Suppose we have a deck of 10 cards - say the ace to ten of spades
    (denoted 1,. . . , 10 for definitiveness). Permutations of the deck may be
    regarded as elements of S(10).

    Define s to be the ‘shuffle’ which hides the top card:
    s = 1 2 3 4 5 6 7 8 9 10
    2 4 6 8 10 1 3 5 7 9 .

    Let t be the ‘shuffe’ that leaves the first card unchanged:
    t = 1 2 3 4 5 6 7 8 9 10
    1 3 5 7 9 2 4 6 8 10.

    Finally, let c be the ‘cut’:

    c = 1 2 3 4 5 6 7 8 9 10
    6 7 8 9 10 1 2 3 4 5 .

    (a) Write s, t, c, cs and scs using cycle notation, as a product of disjoint
    cycles, and find the sign of each.

    (b) For each of these basic and combined shuffes s, t, c, cs and scs, how many times must it be repeated before the cards are returned to their original positions?
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by anncar View Post
    Suppose we have a deck of 10 cards - say the ace to ten of spades
    (denoted 1,. . . , 10 for definitiveness). Permutations of the deck may be
    regarded as elements of S(10).

    Define s to be the ‘shuffle’ which hides the top card:
    s = 1 2 3 4 5 6 7 8 9 10
    2 4 6 8 10 1 3 5 7 9 .

    Let t be the ‘shuffe’ that leaves the first card unchanged:
    t = 1 2 3 4 5 6 7 8 9 10
    1 3 5 7 9 2 4 6 8 10.

    Finally, let c be the ‘cut’:

    c = 1 2 3 4 5 6 7 8 9 10
    6 7 8 9 10 1 2 3 4 5 .

    (a) Write s, t, c, cs and scs using cycle notation, as a product of disjoint
    cycles, and find the sign of each.

    (b) For each of these basic and combined shuffes s, t, c, cs and scs, how many times must it be repeated before the cards are returned to their original positions?

    a)

    s = (1,2,4,8,5,10,9,7,3,6)
    t = (2,3,5,9,8,6)(4,7)
    c = (1,6)(2,7)(3,8)(4,9)(5,10)

     <br />
cs = (1,6)(2,7)(3,8)(4,9)(5,10)(1,2,4,8,5,10,9,7,3,6) <br />

    cs = \left( {\begin{array}{cccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 7 & 9 & 1 & 3 & 5 & 6 & 8 & 10 & 2 & 4 \end{array} } \right) = (1,7,8,10,4,3)(2,9)

    scs = (1,2,4,8,5,10,9,7,3,6)(1,6)(2,7)(3,8)(4,9)(5,10)(1  ,2,4,8,5,10,9,7,3,6)

    scs = \left( {\begin{array}{cccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 3 & 7 & 2 & 6 & 10 & 1 & 5 & 9 & 4 & 8 \end{array} } \right) = (1,3,2,7,5,10,8,9,4,6)

    b) n times where n is the length of the longest cycle..
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