  We apply our geometrical theory for counting placements of $q$ nonattacking
on an $n\times n$ chessboard, from Parts~I and II, to partial queens: that is,
chess pieces with any combination of horizontal, vertical, and
$45^\circ$-diagonal moves. Parts~I and II showed that for any rider (a piece
with moves of unlimited length) the answer will be a quasipolynomial function
of $n$ in which the coefficients are essentially polynomials in $q$. Those
general results gave the three highest-order coefficients of the counting
quasipolynomial and formulas for counting placements of two nonattacking pieces
and the combinatorially distinct types of such placements.
  By contrast, the unified framework we present here for partial queens allows
us to explicitly compute the four highest-order coefficients of the counting
quasipolynomial, show that the five highest-order coefficients are constant
(independent of $n$), and find the period of the next coefficient (which
depends upon the exact set of moves). Furthermore, for three nonattacking
partial queens we are able to prove formulas for the total number of
nonattacking placements and for the number of their combinatorially distinct
types.
  The method of proof, as in the previous parts, is by detailed analysis of the
lattice of subspaces of an inside-out polytope.
