A123320 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.

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

Offset

0,8

Comments

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.

Links

Marc LeBrun, First 100 rows, flattened

Example

T(5,2)=4 because the (5,2) shuffle cycles with period 4:
12345 --> 31425 --> 43215 --> 24135 --> 12345 etc.

Crossrefs

Cf. A002326.

Sequence in context: A039961 A108299 A065941 * A054123 A119269 A363349

Adjacent sequences: A123317 A123318 A123319 * A123321 A123322 A123323

Keyword

easy,nonn,tabl

Author

Marc LeBrun, Sep 25 2006

Status

approved