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

Author

Enoch Haga

Keywords

• easy
• nonn

Examples

			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.)
		

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

• Sean A. Irvine, Java program (github)

Crossrefs

Cf. A024222.