Integer partitions

background

n:=9

md:=5

 

n:=12

md:=7

 

 

n:=6

md:=11

 

 

will be a good choice

 

\(p(5k+4)\equiv 0 \pmod 5\)

\(p(7k+5)\equiv 0 \pmod 7\)

\(p(11k+6)\equiv 0 \pmod {11}\)

 

 

partition and rank

(*define a integer you want to investigate*)
n := 12
(*choose the proper moduli for the partition statistics*)
md := 7
S[n_] := IntegerPartitions[n]
(*define the rank of a partition with the name "pr"*)
pr[s_] := Max[s] - Length[s]
(*modulus distribution partition rank *)
Sort[Table[Mod[pr[s], md], {s, S[n]}]]
(*list of paritions with rank*)
Do[Print[s, ", rank=", pr[s], "\[Congruent]",
  Mod[pr[s], md], "(mod ", md, ")"], {s, S[n]}]
(*you will see p (n),the partition statistics and list of paritions \
with rank*)

 

 

partition and crank

(* Choose n*)
n := 6
Om[s_] := Count[s, 1]
KK[s_] := Select[s, # > Om[s] &]
Mu[s_] := Length[KK[s]]
Crank[s_] := If[Om[s] == 0, Max[s], Mu[s] - Om[s]]
NV[m_, n_] := Length[Select[IntegerPartitions[n], Crank[#] == m &]]
Do[Print[s, ",", Max[s], ",", Om[s], ",", Mu[s], ",", Crank[s]], {s,
  IntegerPartitions[n]}]
(* a partion of "n", maximum part, number of ones=w, number of parts \
larger than w, the crank *)

 

{6},6,0,1,6
{5,1},5,1,1,0
{4,2},4,0,2,4
{4,1,1},4,2,1,-1
{3,3},3,0,2,3
{3,2,1},3,1,2,1
{3,1,1,1},3,3,0,-3
{2,2,2},2,0,3,2
{2,2,1,1},2,2,0,-2
{2,1,1,1,1},2,4,0,-4
{1,1,1,1,1,1},1,6,0,-6

 

various partitions

(* partitions with at most 5 parts *)
IntegerPartitions[7, 5]

(* partition into exactly three parts *)
VS[n_] := IntegerPartitions[n, {3}]
VS[11]

 

(* number of partitions into distinct parts *)
PartitionsQ[11]

 

(* partition into odd parts *)
IntegerPartitions[11, All, {1, 3, 5, 7, 9, 11}]