Integer partitions

background

n:=9

md:=5

 

n:=12

md:=7

 

will be a good choice

 

 

mathematica code

<< Combinatorica`
(*define a integer you want to investigate*)
n := 12
(* choose the proper moduli for the partition statistics *)
md := 7
S[n_] := Partitions[n]
(*define the rank of a partition with the name "pr"*)
pr[s_] := Max[s] - Length[s]
S[n]
PartitionsP[n]
(*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 p (n), the partition statistics and list of paritions \
with rank*)