Let A and B be nonnegative
matrices of orders m and n respectively. In this paper we derive some properties of
the permanent of the direct product A × B of A with B. Specifically we prove
that
with equality if and only if A or B has at most one nonzero term in its permanent
expansion. We also show that every term in the permanent expansion of A × B is
expressible as the product of n terms in the permanent expansion of A and m terms
in the permanent expansion of B, and conversely. This implies that a minimal
positive number Km,n exists such that
for all nonnegative matrices A and B of orders m and n respectively. A conjecture is
given for the value of Km,n.
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