f[n_] := f[Range[n^2], n, Total@Range[n^2]/n]

f[in_, n_, s_] :=
  If[Length@in == n, 1,
    Total[f[Complement[Rest@in, #], n, s] & /@ (
    Select[Subsets[Rest@in, {n - 1}], First@in + Total@# == s &])]]