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# Combination Lock

Consider a combination lock consisting of buttons that can be pressed in any combination (including multiple buttons at once), but in such a way that each number is pressed exactly once. Then the number of possible combination locks with buttons is given by the number of lists (i.e., ordered sets) of disjoint nonempty subsets of the set that contain each number exactly once. For example, there are three possible combination locks with two buttons: , , and . Similarly there are 13 possible three-button combination locks: , , , , , , , , , , , , .

satisfies the linear recurrence equation

 (1)

with . This can also be written

 (2) (3)

where the definition has been used. Furthermore,

 (4) (5)

where are Eulerian numbers. In terms of the Stirling numbers of the second kind ,

 (6)

can also be given in closed form as

 (7)

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

The quantity

 (8)

satisfies the inequality

 (9)

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## References

Sloane, N. J. A. Sequence A000670/M2952 in "The On-Line Encyclopedia of Integer Sequences."Velleman, D. J. and Call, G. S. "Permutations and Combination Locks." Math. Mag. 68, 243-253, 1995.

Combination Lock

## Cite this as:

Weisstein, Eric W. "Combination Lock." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CombinationLock.html