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)
|
