  We derive a bilinear expansion expressing elements of a lattice of KP
$\tau$-functions, labelled by partitions, as a sum over products of pairs of
elements of an associated lattice of BKP $\tau$-functions, labelled by strict
partitions. This generalizes earlier results relating determinants and
Pfaffians of minors of skew symmetric matrices, with applications to Schur
functions and Schur $Q$-functions. It is deduced using the representations of
KP and BKP $\tau$-functions as vacuum expectation values (VEV's) of products of
fermionic operators of charged and neutral type, respectively. The lattice is
generated by insertion of products of pairs of charged creation and
annihilation operators. The result follows from expanding the product as a sum
of monomials in the neutral fermionic generators and applying a factorization
theorem for VEV's of products of operators in the mutually commuting
subalgebras. Applications include the case of inhomogeneous polynomial
$\tau$-functions of KP and BKP type.
