Integer partitions

n := 9
(* choose the proper moduli for the partition statistics *)
md := 5
S[n_] := Partitions[n]
(*define the rank of a partition with the name "pr"*)
pr[s_] := Max[s] - Length[s]
S[9]
(*partition statistics*)
Table[Mod[pr[s], md], {s, S[n]}]
Sort[%]
(*list of paritions with rank*)
Do[Print[s, ", rank=", pr[s], "\[Congruent]",
  Mod[Max[s] - Length[s], md], "(mod ", md, ")"], {s, S[n]}]
(*you will see the partition statistics and list of paritions with \
rank*)