A combination lock considered here consists of 
buttons, where 
is a nonnegative integer.
A code presses every button exactly once, allowing multiple buttons to be pressed
simultaneously. The number 
of possible codes is the number of lists
of disjoint nonempty subsets of the set 
that contain each number exactly once. For example,
there are three possible codes with two buttons: 
, 
, and 
. Similarly there are 13 possible three-button codes:
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
.
The numbers 
are the ordered Bell numbers, also called Fubini
numbers. 
satisfies the linear recurrence equation
 |
(1)
|
for 
,
with 
counting the empty code. For 
, this can also be written
 |  |  |
(2)
|
 |  |  |
(3)
|
where the definition 
has been used. For 
,
 |  |  |
(4)
|
 |  |  |
(5)
|
where 
are Eulerian numbers. In terms of the Stirling
numbers of the second kind 
,
 |
(6)
|
For 
,
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 for 
 |
(9)
|
