The beta invariant of a matroid 
on a finite ground set 
is the integer
 |
where 
is the matroid rank function.
It is nonnegative, and for a matroid with at least two
elements it is positive precisely when 
is connected. For a connected matroid
with at least two elements, it is the coefficient
of 
(and also of 
)
in the Tutte polynomial 
.
