  For every regular graph, we define a sequence of integers, using the
recursion of the Martin polynomial. This sequence counts spanning tree
partitions and constitutes the diagonal coefficients of powers of the Kirchhoff
polynomial. We prove that this sequence respects all known symmetries of
Feynman period integrals in quantum field theory. We show that other quantities
with this property, the $c_2$ invariant and the extended graph permanent, are
essentially determined by our new sequence. This proves the completion
conjecture for the $c_2$ invariant at all primes, and also that it is fixed
under twists. We conjecture that our invariant is perfect: Two Feynman periods
are equal, if and only if, their Martin sequences are equal.
