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Supreme Successive Shuffle (Posted on 2009-12-15) Difficulty: 4 of 5
A deck of M cards is numbered 1 to M and shuffled, and dealt from top to bottom.

Denoting the probability of dealing at least one pair of successive cards in their proper order (that is, a 1 followed by a 2 or, a 2 followed by a 3, and so on) at any position in the deck by s(M), determine s(M) as M → ∞ (The pairs may overlap. For example, for M=5, we have two successive pairs corresponding to 73452.)

As a bonus, what is the expected number of such successive pairs in an M card deck as a function of M?

No Solution Yet Submitted by K Sengupta    
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  Subject Author Date
Solutionre: Some thoughts (Spoiler) FinishedJer2009-12-18 16:37:57
re(3): Recursive two-way formula. Fixed. higher numbersCharlie2009-12-18 13:44:32
re(2): Recursive two-way formula. Fixed.Charlie2009-12-18 12:32:22
Some ThoughtsSome thoughts (Spoiler)Harry2009-12-18 00:18:57
re: Recursive two-way formula. Fixed.Jer2009-12-17 10:16:29
re: Recursive two-way formula.Charlie2009-12-17 03:25:01
Recursive two-way formula.Jer2009-12-16 18:55:43
re: For M=1 to 10 by counting---plus conjectureCharlie2009-12-15 12:12:20
Some ThoughtsFor M=1 to 10 by counting---plus conjectureCharlie2009-12-15 11:52:44
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