자연수의 분할(partition)과 rank/crank 목록

 

 

개요

• rank of a partition= the largest part of the partition minus the number of parts

 

목록

 

 

{{2},{1,1}}

 

{{3},{2,1},{1,1,1}}

 

{{4},{3,1},{2,2},{2,1,1},{1,1,1,1}}

 

{{5},{4,1},{3,2},{3,1,1},{2,2,1},{2,1,1,1},{1,1,1,1,1}}

 

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

 

{{7},{6,1},{5,2},{5,1,1},{4,3},{4,2,1},{4,1,1,1},{3,3,1},{3,2,2},{3,2,1,1},{3,1,1,1,1},{2,2,2,1},{2,2,1,1,1},{2,1,1,1,1,1},{1,1,1,1,1,1,1}}

 

{{8},{7,1},{6,2},{6,1,1},{5,3},{5,2,1},{5,1,1,1},{4,4},{4,3,1},{4,2,2},{4,2,1,1},{4,1,1,1,1},{3,3,2},{3,3,1,1},{3,2,2,1},{3,2,1,1,1},{3,1,1,1,1,1},{2,2,2,2},{2,2,2,1,1},{2,2,1,1,1,1},{2,1,1,1,1,1,1},{1,1,1,1,1,1,1,1}}

 

9의 분할

{{9},{8,1},{7,2},{7,1,1},{6,3},{6,2,1},{6,1,1,1},{5,4},{5,3,1},{5,2,2},{5,2,1,1},{5,1,1,1,1},{4,4,1},{4,3,2},{4,3,1,1},{4,2,2,1},{4,2,1,1,1},{4,1,1,1,1,1},{3,3,3},{3,3,2,1},{3,3,1,1,1},{3,2,2,2},{3,2,2,1,1},{3,2,1,1,1,1},{3,1,1,1,1,1,1},{2,2,2,2,1},{2,2,2,1,1,1},{2,2,1,1,1,1,1},{2,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1}}

 

{9}, rank=8≡3 (mod 5)
{8,1}, rank=6≡1 (mod 5)
{7,2}, rank=5≡0 (mod 5)
{7,1,1}, rank=4≡4 (mod 5)

{6,3}, rank=4≡4 (mod 5)

{6,2,1}, rank=3≡3 (mod 5)
{6,1,1,1}, rank=2≡2 (mod 5)

{5,4}, rank=3≡3 (mod 5)

{5,3,1}, rank=2≡2 (mod 5)
{5,2,2}, rank=2≡2 (mod 5)
{5,2,1,1}, rank=1≡1 (mod 5)
{5,1,1,1,1}, rank=0≡0 (mod 5)
{4,4,1}, rank=1≡1 (mod 5)
{4,3,2}, rank=1≡1 (mod 5)
{4,3,1,1}, rank=0≡0 (mod 5)
{4,2,2,1}, rank=0≡0 (mod 5)
{4,2,1,1,1}, rank=-1≡4 (mod 5)

{4,1,1,1,1,1}, rank=-2≡3 (mod 5)

{3,3,3}, rank=0≡0 (mod 5)
{3,3,2,1}, rank=-1≡4 (mod 5)

{3,3,1,1,1}, rank=-2≡3 (mod 5)

{3,2,2,2}, rank=-1≡4 (mod 5)

{3,2,2,1,1}, rank=-2≡3 (mod 5)

{3,2,1,1,1,1}, rank=-3≡2 (mod 5)
{3,1,1,1,1,1,1}, rank=-4≡1 (mod 5)
{2,2,2,2,1}, rank=-3≡2 (mod 5)
{2,2,2,1,1,1}, rank=-4≡1 (mod 5)
{2,2,1,1,1,1,1}, rank=-5≡0 (mod 5)
{2,1,1,1,1,1,1,1}, rank=-6≡4 (mod 5)

{1,1,1,1,1,1,1,1,1}, rank=-8≡2 (mod 5)

 

10의 분할

{{10},{9,1},{8,2},{8,1,1},{7,3},{7,2,1},{7,1,1,1},{6,4},{6,3,1},{6,2,2},{6,2,1,1},{6,1,1,1,1},{5,5},{5,4,1},{5,3,2},{5,3,1,1},{5,2,2,1},{5,2,1,1,1},{5,1,1,1,1,1},{4,4,2},{4,4,1,1},{4,3,3},{4,3,2,1},{4,3,1,1,1},{4,2,2,2},{4,2,2,1,1},{4,2,1,1,1,1},{4,1,1,1,1,1,1},{3,3,3,1},{3,3,2,2},{3,3,2,1,1},{3,3,1,1,1,1},{3,2,2,2,1},{3,2,2,1,1,1},{3,2,1,1,1,1,1},{3,1,1,1,1,1,1,1},{2,2,2,2,2},{2,2,2,2,1,1},{2,2,2,1,1,1,1},{2,2,1,1,1,1,1,1},{2,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1}}

 

{{11},{10,1},{9,2},{9,1,1},{8,3},{8,2,1},{8,1,1,1},{7,4},{7,3,1},{7,2,2},{7,2,1,1},{7,1,1,1,1},{6,5},{6,4,1},{6,3,2},{6,3,1,1},{6,2,2,1},{6,2,1,1,1},{6,1,1,1,1,1},{5,5,1},{5,4,2},{5,4,1,1},{5,3,3},{5,3,2,1},{5,3,1,1,1},{5,2,2,2},{5,2,2,1,1},{5,2,1,1,1,1},{5,1,1,1,1,1,1},{4,4,3},{4,4,2,1},{4,4,1,1,1},{4,3,3,1},{4,3,2,2},{4,3,2,1,1},{4,3,1,1,1,1},{4,2,2,2,1},{4,2,2,1,1,1},{4,2,1,1,1,1,1},{4,1,1,1,1,1,1,1},{3,3,3,2},{3,3,3,1,1},{3,3,2,2,1},{3,3,2,1,1,1},{3,3,1,1,1,1,1},{3,2,2,2,2},{3,2,2,2,1,1},{3,2,2,1,1,1,1},{3,2,1,1,1,1,1,1},{3,1,1,1,1,1,1,1,1},{2,2,2,2,2,1},{2,2,2,2,1,1,1},{2,2,2,1,1,1,1,1},{2,2,1,1,1,1,1,1,1},{2,1,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1,1}}

 

{{12},{11,1},{10,2},{10,1,1},{9,3},{9,2,1},{9,1,1,1},{8,4},{8,3,1},{8,2,2},{8,2,1,1},{8,1,1,1,1},{7,5},{7,4,1},{7,3,2},{7,3,1,1},{7,2,2,1},{7,2,1,1,1},{7,1,1,1,1,1},{6,6},{6,5,1},{6,4,2},{6,4,1,1},{6,3,3},{6,3,2,1},{6,3,1,1,1},{6,2,2,2},{6,2,2,1,1},{6,2,1,1,1,1},{6,1,1,1,1,1,1},{5,5,2},{5,5,1,1},{5,4,3},{5,4,2,1},{5,4,1,1,1},{5,3,3,1},{5,3,2,2},{5,3,2,1,1},{5,3,1,1,1,1},{5,2,2,2,1},{5,2,2,1,1,1},{5,2,1,1,1,1,1},{5,1,1,1,1,1,1,1},{4,4,4},{4,4,3,1},{4,4,2,2},{4,4,2,1,1},{4,4,1,1,1,1},{4,3,3,2},{4,3,3,1,1},{4,3,2,2,1},{4,3,2,1,1,1},{4,3,1,1,1,1,1},{4,2,2,2,2},{4,2,2,2,1,1},{4,2,2,1,1,1,1},{4,2,1,1,1,1,1,1},{4,1,1,1,1,1,1,1,1},{3,3,3,3},{3,3,3,2,1},{3,3,3,1,1,1},{3,3,2,2,2},{3,3,2,2,1,1},{3,3,2,1,1,1,1},{3,3,1,1,1,1,1,1},{3,2,2,2,2,1},{3,2,2,2,1,1,1},{3,2,2,1,1,1,1,1},{3,2,1,1,1,1,1,1,1},{3,1,1,1,1,1,1,1,1,1},{2,2,2,2,2,2},{2,2,2,2,2,1,1},{2,2,2,2,1,1,1,1},{2,2,2,1,1,1,1,1,1},{2,2,1,1,1,1,1,1,1,1},{2,1,1,1,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1,1,1}}

 

 

12의 분할

{12}, rank=11\[Congruent]4(mod 7)
{11,1}, rank=9\[Congruent]2(mod 7)
{10,2}, rank=8\[Congruent]1(mod 7)
{10,1,1}, rank=7\[Congruent]0(mod 7)
{9,3}, rank=7\[Congruent]0(mod 7)
{9,2,1}, rank=6\[Congruent]6(mod 7)
{9,1,1,1}, rank=5\[Congruent]5(mod 7)
{8,4}, rank=6\[Congruent]6(mod 7)
{8,3,1}, rank=5\[Congruent]5(mod 7)
{8,2,2}, rank=5\[Congruent]5(mod 7)
{8,2,1,1}, rank=4\[Congruent]4(mod 7)
{8,1,1,1,1}, rank=3\[Congruent]3(mod 7)
{7,5}, rank=5\[Congruent]5(mod 7)
{7,4,1}, rank=4\[Congruent]4(mod 7)
{7,3,2}, rank=4\[Congruent]4(mod 7)
{7,3,1,1}, rank=3\[Congruent]3(mod 7)
{7,2,2,1}, rank=3\[Congruent]3(mod 7)
{7,2,1,1,1}, rank=2\[Congruent]2(mod 7)
{7,1,1,1,1,1}, rank=1\[Congruent]1(mod 7)
{6,6}, rank=4\[Congruent]4(mod 7)
{6,5,1}, rank=3\[Congruent]3(mod 7)
{6,4,2}, rank=3\[Congruent]3(mod 7)
{6,4,1,1}, rank=2\[Congruent]2(mod 7)
{6,3,3}, rank=3\[Congruent]3(mod 7)
{6,3,2,1}, rank=2\[Congruent]2(mod 7)
{6,3,1,1,1}, rank=1\[Congruent]1(mod 7)
{6,2,2,2}, rank=2\[Congruent]2(mod 7)
{6,2,2,1,1}, rank=1\[Congruent]1(mod 7)
{6,2,1,1,1,1}, rank=0\[Congruent]0(mod 7)
{6,1,1,1,1,1,1}, rank=-1\[Congruent]6(mod 7)
{5,5,2}, rank=2\[Congruent]2(mod 7)
{5,5,1,1}, rank=1\[Congruent]1(mod 7)
{5,4,3}, rank=2\[Congruent]2(mod 7)
{5,4,2,1}, rank=1\[Congruent]1(mod 7)
{5,4,1,1,1}, rank=0\[Congruent]0(mod 7)
{5,3,3,1}, rank=1\[Congruent]1(mod 7)
{5,3,2,2}, rank=1\[Congruent]1(mod 7)
{5,3,2,1,1}, rank=0\[Congruent]0(mod 7)
{5,3,1,1,1,1}, rank=-1\[Congruent]6(mod 7)
{5,2,2,2,1}, rank=0\[Congruent]0(mod 7)
{5,2,2,1,1,1}, rank=-1\[Congruent]6(mod 7)
{5,2,1,1,1,1,1}, rank=-2\[Congruent]5(mod 7)
{5,1,1,1,1,1,1,1}, rank=-3\[Congruent]4(mod 7)
{4,4,4}, rank=1\[Congruent]1(mod 7)
{4,4,3,1}, rank=0\[Congruent]0(mod 7)
{4,4,2,2}, rank=0\[Congruent]0(mod 7)
{4,4,2,1,1}, rank=-1\[Congruent]6(mod 7)
{4,4,1,1,1,1}, rank=-2\[Congruent]5(mod 7)
{4,3,3,2}, rank=0\[Congruent]0(mod 7)
{4,3,3,1,1}, rank=-1\[Congruent]6(mod 7)
{4,3,2,2,1}, rank=-1\[Congruent]6(mod 7)
{4,3,2,1,1,1}, rank=-2\[Congruent]5(mod 7)
{4,3,1,1,1,1,1}, rank=-3\[Congruent]4(mod 7)
{4,2,2,2,2}, rank=-1\[Congruent]6(mod 7)
{4,2,2,2,1,1}, rank=-2\[Congruent]5(mod 7)
{4,2,2,1,1,1,1}, rank=-3\[Congruent]4(mod 7)
{4,2,1,1,1,1,1,1}, rank=-4\[Congruent]3(mod 7)
{4,1,1,1,1,1,1,1,1}, rank=-5\[Congruent]2(mod 7)
{3,3,3,3}, rank=-1\[Congruent]6(mod 7)
{3,3,3,2,1}, rank=-2\[Congruent]5(mod 7)
{3,3,3,1,1,1}, rank=-3\[Congruent]4(mod 7)
{3,3,2,2,2}, rank=-2\[Congruent]5(mod 7)
{3,3,2,2,1,1}, rank=-3\[Congruent]4(mod 7)
{3,3,2,1,1,1,1}, rank=-4\[Congruent]3(mod 7)
{3,3,1,1,1,1,1,1}, rank=-5\[Congruent]2(mod 7)
{3,2,2,2,2,1}, rank=-3\[Congruent]4(mod 7)
{3,2,2,2,1,1,1}, rank=-4\[Congruent]3(mod 7)
{3,2,2,1,1,1,1,1}, rank=-5\[Congruent]2(mod 7)
{3,2,1,1,1,1,1,1,1}, rank=-6\[Congruent]1(mod 7)
{3,1,1,1,1,1,1,1,1,1}, rank=-7\[Congruent]0(mod 7)
{2,2,2,2,2,2}, rank=-4\[Congruent]3(mod 7)
{2,2,2,2,2,1,1}, rank=-5\[Congruent]2(mod 7)
{2,2,2,2,1,1,1,1}, rank=-6\[Congruent]1(mod 7)
{2,2,2,1,1,1,1,1,1}, rank=-7\[Congruent]0(mod 7)
{2,2,1,1,1,1,1,1,1,1}, rank=-8\[Congruent]6(mod 7)
{2,1,1,1,1,1,1,1,1,1,1}, rank=-9\[Congruent]5(mod 7)
{1,1,1,1,1,1,1,1,1,1,1,1}, rank=-11\[Congruent]3(mod 7)

 

 

 

재미있는 사실

 

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역사

• 수학사연표

 

 

메모

Ramanujan's congruences and Dyson's crank
George E. Andrews*† and Ken Ono‡

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1266147/

 

 

 

수학용어번역

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사전 형태의 자료

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• http://www.wolframalpha.com/input/?i=
• NIST Digital Library of Mathematical Functions
• The On-Line Encyclopedia of Integer Sequences
  
  • http://www.research.att.com/~njas/sequences/?q=

 

 

관련논문

• http://www.jstor.org/action/doBasicSearch?Query=
• http://dx.doi.org/

 

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