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A123320
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Table of the cycle lengths for "imperfect" (generalized) faro shuffles with cut of size k returning a deck of size n to its original order.
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2
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 4, 2, 1, 1, 1, 5, 6, 4, 2, 1, 1, 1, 6, 6, 3, 4, 2, 1, 1, 1, 7, 12, 10, 3, 4, 2, 1, 1, 1, 8, 8, 6, 6, 3, 4, 2, 1, 1, 1, 9, 20, 20, 9, 6, 3, 4, 2, 1, 1, 1, 10, 10, 21, 8, 10, 6, 3, 4, 2, 1, 1, 1, 11, 30, 24, 20, 11, 10, 6, 3, 4, 2, 1, 1, 1, 12, 12, 35, 9, 12
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OFFSET
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0,8
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COMMENTS
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An "imperfect" (generalized) faro shuffle with cut of size k for a deck of size n is performed by first cutting the deck into a top pile of k cards and a bottom pile of n-k cards, performing a perfect faro shuffle on the bottomost min(k,n-k) cards of each pile and placing any remaining cards on top of the deck. (Thus k may range from 0 to n inclusive, hence the offset is 0). The central column T(2k,k) gives the "perfect" faro shuffle cycles A002326.
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LINKS
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EXAMPLE
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T(5,2)=4 because the (5,2) shuffle cycles with period 4:
12345 --> 31425 --> 43215 --> 24135 --> 12345 etc.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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