_Marc LeBrun (mlb(AT)well.com), _, Sep 25 2006

Marc LeBrun, <a href="/A123320/b123320.txt">First 100 rows, flattened</a>

easy,nonn,tabl,new

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.

Marc LeBrun, <a href="http://www.research.att.com/~njas/sequences/b123320.txt">First 100 rows, flattened</a>

easy,nonn,tabl,new

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

0,8

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.

Marc LeBrun, <a href="http://www.research.att.com/~njas/sequences/b123320.txt">First 100 rows, flattened</a>

T(5,2)=4 because the (5,2) shuffle cycles with period 4:

12345 --> 31425 --> 43215 --> 24135 --> 12345 etc.

Cf. A002326.

easy,nonn,tabl,new

Marc LeBrun (mlb(AT)well.com), Sep 25 2006

approved