Remove[a, c, s];
a[n_Integer] :=
  a[n] =
    Flatten[
      Table[F[i, j, k, l], {l, 1, n}, {k, 1, l - 1}, {j, 1, k - 1}, {i, 1, j - 1}]]
        /. F[i_, j_, k_, l_] -> {i, j, k, l};
c = {1,2,3,4};
s[n_Integer] :=
  Sum[(-1)^Length[Intersection[a[n][[u]], c]], {u, 1, Length[a[n]]}];