## Combination Lock

Let a combination of buttons be a Sequence of disjoint nonempty Subsets of the Set . If the number of possible combinations is denoted , then satisfies the Recurrence Relation

 (1)

with . This can also be written
 (2)

where the definition has been used. Furthermore,
 (3)

where are Eulerian Numbers. In terms of the Stirling Numbers of the Second Kind ,
 (4)

can also be given in closed form as
 (5)

where is the Polylogarithm. The first few values of for , 2, ... are 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, ... (Sloane's A000670).

The quantity

 (6)

satisfies the inequality
 (7)

References

Sloane, N. J. A. Sequence A000670/M2952 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Velleman, D. J. and Call, G. S. Permutations and Combination Locks.'' Math. Mag. 68, 243-253, 1995.