  The values of the normalized homogeneous weight are determined for arbitrary
finite Frobenius rings and expressed in a form that is independent from a
generating character and the M\"obius function on the ring. The weight
naturally induces a partition of the ring, which is invariant under left or
right multiplication by units. It is shown that the character-theoretic
left-sided dual of this partition coincides with the right-sided dual, and even
more, the left- and right-sided Krawtchouk coefficients coincide. An example is
provided showing that this is not the case for general invariant partitions if
the ring is not semisimple.
