A minimal cover is a cover for which removal of any single member destroys the covering property. For example, of the five covers
of 
, namely 
, 
, 
, 
, and 
, only 
and 
are minimal covers.
Similarly, the minimal covers of 
are given by 
, 
, 
, 
, 
, 
, 
, and 
. The numbers of minimal covers of a fixed labeled
-element set,
with unordered cover members, for 
, 2, ..., are 1, 2, 8, 49, 462, 6424, 129425, ... (OEIS A046165).
Royle (2000) proved that there is a one-to-one correspondence between split graphs on 
vertices, considered up to graph
isomorphism, and minimal covers of an 
-element set up to permutations
of its elements.
Let 
be the number of minimal covers of 
with 
members, for positive integers 
and 
. Then
 |
where 
is a binomial coefficient, 
is a Stirling
number of the second kind, and
 |
Special cases include 
and 
.
The table below gives a triangle of 
(OEIS A035348).
