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 rank and crank

(*define a integer you want to investigate*)n := 6
(*choose the proper moduli for the partition statistics*)
md := 2
S[n_] := IntegerPartitions[n]
(*define the rank of a partition with the name "pr"*)
pr[s_] := Max[s] - Length[s]
(*define the crank of a partition with the name "crank"*)
Om[s_] := Count[s, 1]
Mu[s_] := Length[Select[s, # > Om[s] &]]
crank[s_] := If[Om[s] == 0, Max[s], Mu[s] - Om[s]]
(*modulus distribution of partition rank*)
Sort[Tally[Table[Mod[pr[s], md], {s, S[n]}]]]
(*modulus distribution of partition crank*)
Sort[Tally[Table[Mod[crank[s], md], {s, S[n]}]]]
(*list of paritions with rank& crank*)
Do[Print[s, ", rank=", pr[s], "\[Congruent]", Mod[pr[s], md], "(mod ",
   md, ")", ", crank=", crank[s], "\[Congruent]", Mod[crank[s], md],
  "(mod ", md, ")"], {s, S[n]}]
(*you will see the distribution of rank/crank modulus,the partition \
statistics and list of paritions with rank&crank*)

 

 

number of partitions with odd and even rank

• for theoretical background, see rank of partition and mock theta function

S[n_] := IntegerPartitions[n]
pr[s_] := Max[s] - Length[s]
PrOd[n_] := Length[Select[S[n], OddQ[pr[#]] &]]
PrEv[n_] := Length[Select[S[n], EvenQ[pr[#]] &]]
alpha[n_] := PrEv[n] - PrOd[n]
Table[alpha[n], {n, 1, 20}]

 

 

• the generating function is can be shown by
  Series[(1 + 4 Sum[(-1)^n q^(n (3 n + 1)/2)/(1 + q^n), {n, 1, 10}])/Sum[(-1)^n q^(n (3 n + 1)/2), {n, -8, 8}], {q, 0, 100}]
  

 

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}]