"Integer partitions"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) |
|||
| (사용자 3명의 중간 판 12개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
| − | + | ==background== | |
n:=9 | n:=9 | ||
| 5번째 줄: | 5번째 줄: | ||
md:=5 | md:=5 | ||
| − | + | ||
n:=12 | n:=12 | ||
| 11번째 줄: | 11번째 줄: | ||
md:=7 | md:=7 | ||
| − | + | ||
| − | + | ||
n:=6 | n:=6 | ||
| 19번째 줄: | 19번째 줄: | ||
md:=11 | md:=11 | ||
| − | + | ||
| − | + | ||
will be a good choice | will be a good choice | ||
| − | + | ||
<math>p(5k+4)\equiv 0 \pmod 5</math> | <math>p(5k+4)\equiv 0 \pmod 5</math> | ||
| 33번째 줄: | 33번째 줄: | ||
<math>p(11k+6)\equiv 0 \pmod {11}</math> | <math>p(11k+6)\equiv 0 \pmod {11}</math> | ||
| − | + | ||
| − | + | ||
| − | + | ==partition rank and crank== | |
| − | (*define a integer you want to investigate*) | + | (*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 conjecture|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] | |
| − | + | ||
| − | (* partition into odd parts *) | + | (* number of partitions into distinct parts *) PartitionsQ[11] |
| + | |||
| + | |||
| + | |||
| + | (* partition into odd parts *) IntegerPartitions[11, All, {1, 3, 5, 7, 9, 11}] | ||
| + | [[분류:math and physics]] | ||
| + | [[분류:migrate]] | ||
2020년 12월 28일 (월) 05:18 기준 최신판
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}]