A024542 Lower bound on cumulative mean of distances between cards after n shuffles in A024222.

0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 7, 7, 8, 8, 11, 11, 12, 12, 12, 12, 12, 12, 11, 12, 14, 14, 15, 15, 16, 16, 14, 14, 17, 18, 17, 17, 15, 15, 19, 20, 20, 20, 13, 13, 13, 14, 22, 22, 22, 22, 23, 23, 18, 18, 24, 24, 25, 25, 26, 26

Offset

1,6

References

Martin Gardner, "Card Shuffles," Mathematical Carnival, chapter 10, pages 123-138. New York: Vintage Books, 1977.

Tim Folger, "Shuffling Into Hyperspace," Discover, 1991 (vol 12, no 1), pages 66-67.

Links

Table of n, a(n) for n=1..80.

Sean A. Irvine, Java program (github)

Example

a(6)=2. Without cut, after shuffle 1 (s1), order is 4 1 5 2 6 3; after s2, 2 4 6 1 3 5; after s3, 1 2 3 4 5 6. Mean distances between cards after s1 = 3.4 (17/5), s2 = 2.6 (13/5), s3 = 1 (5/5). 35/15=2.3, cumulative mean distance, so lower bound is 2. (Cuts change number of shuffles required to return cards to original order, in this case from 3 to 4.)

Crossrefs

Cf. A024222.

Sequence in context: A333535 A364678 A175387 * A355880 A209082 A257684

Adjacent sequences: A024539 A024540 A024541 * A024543 A024544 A024545

Keyword

easy,nonn

Author

Enoch Haga

Status

approved