  We study lattice configurations related to S_n, the scalar product of an
off-shell state and an on-shell state in rational A_n integrable vertex models,
n = {1, 2}. The lattice lines are colourless and oriented. The state variables
are n conserved colours that flow along the line orientations, but do not
necessarily cover every bond in the lattice.
  Choosing boundary conditions such that the positions where the colours flow
into the lattice are fixed, and where they flow out are summed over, we show
that the partition functions of these configurations, with these boundary
conditions, are n-independent. Our results extend to trigonometric A_n models,
and to all n.
  This n-independence explains, in vertex-model terms, results from recent
studies of S_2 [1, 2]. Namely, 1. S_2 which depends on two sets of Bethe roots,
b_1 and b_2, and cannot (as far as we know) be expressed in single determinant
form, degenerates in the limit b_1 -> infinity, and/or b_2 -> infinity, into a
product of determinants, 2. Each of the latter determinants is an A_1
vertex-model partition function.
