The ordered Bell number 
, also called the Fubini number, counts the lists
of disjoint nonempty subsets whose union is a fixed
-element
set. Here 
is a nonnegative integer,
and 
counts the empty list. Unlike the Bell
numbers, these numbers distinguish different orders of the blocks.
For 
,
1, ..., the first values are 1, 1, 3, 13, 75, 541, 4683, 47293, ... (OEIS A000670).
A set partition with 
blocks has 
possible block orders, giving
 |
(1)
|
where 
is a Stirling number of the second
kind. As an identity of formal power series,
and analytically for 
,
the exponential generating function
is
 |
(2)
|
Choosing the first nonempty block gives the recurrence relation
 |
(3)
|
for 
,
where 
is a binomial coefficient.
These numbers count the button codes in the combination lock model, where each block consists of the buttons pressed simultaneously and the block order specifies the sequence of presses (Velleman and Call 1995).
