The polymatroid Tutte polynomial is an activity generating polynomial for the integer bases of an integral polymatroid.
Fix a total order on its ground set 
, let 
be its set of integer bases, and let 
denote the coordinate unit vector
at 
.
For a basis 
,
an element 
is internally active if 
for every 
, and externally active if 
for every 
. Writing 
, 
, and 
for the numbers of only internally active, only externally
active, and simultaneously active elements gives
 |
The polynomial is independent of the chosen total order (Tóthmérész 2026).
For a matroid of rank 
on 
elements, this normalization is related to its ordinary Tutte polynomial 
by
 |
The right side is interpreted as the polynomial obtained after cancellation, including when 
or 
is zero. For example, one element with one integer basis gives
.
Tóthmérész (2026) proves that computation-tree activities, using either a minimum or a maximum rule, also yield this polynomial. The hypergraph Tutte polynomial is a special case using the integer bases supplied by a connected hypergraph.
