"자연수의 분할(partition)과 rank/crank 목록"의 두 판 사이의 차이

수학노트
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==재미있는 사실==
 
==재미있는 사실==
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* In 1944, the crank was first hinted at by Freeman Dyson (2), then an undergraduate at Cambridge University. He had written an article, titled Some Guesses in the Theory of Partitions, for Eureka, the undergraduate mathematics journal of Cambridge.
 
* In 1944, the crank was first hinted at by Freeman Dyson (2), then an undergraduate at Cambridge University. He had written an article, titled Some Guesses in the Theory of Partitions, for Eureka, the undergraduate mathematics journal of Cambridge.
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*  네이버 지식인<br>
 
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** http://kin.search.naver.com/search.naver?where=kin_qna&query=
 
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==[http://kin.search.naver.com/search.naver?where=kin_qna&query=%EC%97%AD%EC%82%AC%EC%88%98%ED%95%99%EC%82%AC%EC%97%B0%ED%91%9C%EB%A9%94%EB%AA%A8 역사]==
 
  
 
* [[수학사 연표]]
 
* [[수학사 연표]]
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==사전 형태의 자료==
 
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* http://ko.wikipedia.org/wiki/
 
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* http://en.wikipedia.org/wiki/
 
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* http://www.wolframalpha.com/input/?i=
 
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
 
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** http://www.research.att.com/~njas/sequences/?q=
 
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==관련논문==
 
==관련논문==
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* Dousse, Jehanne, and Michael H. Mertens. “Asymptotic Formulae for Partition Ranks.” arXiv:1406.6848 [math], June 26, 2014. http://arxiv.org/abs/1406.6848.
 
* [http://dx.doi.org/10.1073/pnas.0507844102 Ramanujan's congruences and Dyson's crank]<br>
 
* [http://dx.doi.org/10.1073/pnas.0507844102 Ramanujan's congruences and Dyson's crank]<br>
 
** George E. Andrews and Ken Ono, PNAS October 25, 2005 vol. 102 no. 43 15277
 
** George E. Andrews and Ken Ono, PNAS October 25, 2005 vol. 102 no. 43 15277
 
* [http://projecteuclid.org/euclid.bams/1183554533 Dyson's crank of a partition]<br>
 
* [http://projecteuclid.org/euclid.bams/1183554533 Dyson's crank of a partition]<br>
 
** George E. Andrews and F. G. Garvan, Bull. Amer. Math. Soc. (N.S.) Volume 18, Number 2 (1988), 167-171
 
** George E. Andrews and F. G. Garvan, Bull. Amer. Math. Soc. (N.S.) Volume 18, Number 2 (1988), 167-171
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* http://www.jstor.org/action/doBasicSearch?Query=
 
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* http://dx.doi.org/10.1073/pnas.0507844102.
 
  
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[[분류:q-급수]]
 
[[분류:q-급수]]
 
[[분류:목록]]
 
[[분류:목록]]
 
[[분류:분할수]]
 
[[분류:분할수]]

2014년 6월 26일 (목) 17:42 판

개요

  • 분할의 rank = 분할에서 가장 큰 수 - 분할의 크기
  • 예
    • 9의 분할인 {7,1,1}의 경우, rank=7-3=4
    • 9의 분할인 {4,3,1,1}의 경우, rank=4-4=0
  • 분할의 crank
    • 분할에서 가장 큰 수 (1이 포함되지 않는 분할의 경우)
    • 분할에서 "1의개수"보다 큰 수 - 1의 개수 (1이 포함되는 경우)
  • 예
    • 9의 분할인 {7,1,1}의 경우, crank=1-2=-1
    • 9의 분할인 {4,3,1,1}의 경우, crank=2-2=0
  • 200까지의 분할수 목록 항목 참조

 

 

목록

  • 분할수와 분할의 목록
  • 경우에 따라 분할에 따른 rank

 

 

1의 분할

  • 분할수 = 1
  • {{1}}

 

2의 분할

  • 분할수 = 2
  • {{2},{1,1}}

 

3의 분할

  • 분할수 = 3
  • {{3},{2,1},{1,1,1}}

 

 

4의 분할

  • 분할수 = 5
  • {{4},{3,1},{2,2},{2,1,1},{1,1,1,1}}

 

 

5의 분할

  • 분할수 = 7
  • {{5},{4,1},{3,2},{3,1,1},{2,2,1},{2,1,1,1},{1,1,1,1,1}}

 

 

6의 분할

  • 분할수 = 11
  • {{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}}
  • 분할의 rank 
    {5, 3, 2, 1, 1, 0, -1, -1, -2, -3, -5}≡ {5, 3, 2, 1, 1, 0, 10, 10, 9, 8, 6} (mod 11)
    rank의 나머지에 7이 없고, 10이 두개
  • 분할의 crank
    {6, 0, 4, -1, 3, 1, -3, 2, -2, -4, -6} ≡ {6, 0, 4, 10, 3, 1, 8, 2, 9, 7, 5} (mod 11)
    crank의 나머지는 고르게 분포되어 있음
  • 분할과 rank, rank (mod 11), crank (mod 11)

{6}, rank=5≡5(mod 11), crank=6≡6(mod 11)
{5,1}, rank=3≡3(mod 11), crank=0≡0(mod 11)
{4,2}, rank=2≡2(mod 11), crank=4≡4(mod 11)
{4,1,1}, rank=1≡1(mod 11), crank=-1≡10(mod 11)
{3,3}, rank=1≡1(mod 11), crank=3≡3(mod 11)
{3,2,1}, rank=0≡0(mod 11), crank=1≡1(mod 11)
{3,1,1,1}, rank=-1≡10(mod 11), crank=-3≡8(mod 11)
{2,2,2}, rank=-1≡10(mod 11), crank=2≡2(mod 11)
{2,2,1,1}, rank=-2≡9(mod 11), crank=-2≡9(mod 11)
{2,1,1,1,1}, rank=-3≡8(mod 11), crank=-4≡7(mod 11)
{1,1,1,1,1,1}, rank=-5≡6(mod 11), crank=-6≡5(mod 11)
{4,3,1,1}, rank=0≡0(mod 5), crank=0≡0(mod 5)
{4,2,2,1}, rank=0≡0(mod 5), crank=2≡2(mod 5)
{4,2,1,1,1}, rank=-1≡4(mod 5), crank=-2≡3(mod 5)
{4,1,1,1,1,1}, rank=-2≡3(mod 5), crank=-5≡0(mod 5)
{3,3,3}, rank=0≡0(mod 5), crank=3≡3(mod 5)
{3,3,2,1}, rank=-1≡4(mod 5), crank=2≡2(mod 5)
{3,3,1,1,1}, rank=-2≡3(mod 5), crank=-3≡2(mod 5)
{3,2,2,2}, rank=-1≡4(mod 5), crank=3≡3(mod 5)
{3,2,2,1,1}, rank=-2≡3(mod 5), crank=-1≡4(mod 5)
{3,2,1,1,1,1}, rank=-3≡2(mod 5), crank=-4≡1(mod 5)
{3,1,1,1,1,1,1}, rank=-4≡1(mod 5), crank=-6≡4(mod 5)
{2,2,2,2,1}, rank=-3≡2(mod 5), crank=3≡3(mod 5)
{2,2,2,1,1,1}, rank=-4≡1(mod 5), crank=-3≡2(mod 5)
{2,2,1,1,1,1,1}, rank=-5≡0(mod 5), crank=-5≡0(mod 5)
{2,1,1,1,1,1,1,1}, rank=-6≡4(mod 5), crank=-7≡3(mod 5)
{1,1,1,1,1,1,1,1,1}, rank=-8≡2(mod 5), crank=-9≡1(mod 5)   

 

 

7의 분할

  • 분할수 = 15
  • {{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}}

 

 

8의 분할

  • 분할수 = 22
  • {{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

  • 분할수 = 30
  • {{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}}
  • 분할의 rank 
    {8,6,5,4,4,3,2,3,2,2,1,0,1,1,0,0,-1,-2,0,-1,-2,-1,-2,-3,-4,-3,-4,-5,-6,-8}
  • 분할의 crank
    {9,0,7,-1,6,1,-2,5,1,5,-1,-3,1,4,0,2,-2,-5,3,2,-3,3,-1,-4,-6,3,-3,-5,-7,-9}
  • 분할과 rank, rank (mod 5), crank (mod 5)

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

 

 

10의 분할

  • 분할수 = 42
  • {{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}}

 

 

11의 분할

  • 분할수 = 56
  • {{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}}

 

 

12의 분할

  • 분할수 = 77
  • {{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}}
  • 분할의 rank
    {11,9,8,7,7,6,5,6,5,5,4,3,5,4,4,3,3,2,1,4,3,3,2,3,2,1,2,1,0,-1,2,1,2,1,0,1,1,0,-1,0,-1,-2,-3,1,0,0,-1,-2,0,-1,-1,-2,-3,-1,-2,-3,-4,-5,-1,-2,-3,-2,-3,-4,-5,-3,-4,-5,-6,-7,-4,-5,-6,-7,-8,-9,-11}
  • 분할과 rank, rank (mod 7)
    {12}, rank=11≡4 (mod 7)
    {11,1}, rank=9≡2 (mod 7)
    {10,2}, rank=8≡1 (mod 7)
    {10,1,1}, rank=7≡0 (mod 7)
    {9,3}, rank=7≡0 (mod 7)
    {9,2,1}, rank=6≡6 (mod 7)
    {9,1,1,1}, rank=5≡5 (mod 7)
    {8,4}, rank=6≡6 (mod 7)
    {8,3,1}, rank=5≡5 (mod 7)
    {8,2,2}, rank=5≡5 (mod 7)
    {8,2,1,1}, rank=4≡4 (mod 7)
    {8,1,1,1,1}, rank=3≡3 (mod 7)
    {7,5}, rank=5≡5 (mod 7)
    {7,4,1}, rank=4≡4 (mod 7)
    {7,3,2}, rank=4≡4 (mod 7)
    {7,3,1,1}, rank=3≡3 (mod 7)
    {7,2,2,1}, rank=3≡3 (mod 7)
    {7,2,1,1,1}, rank=2≡2 (mod 7)
    {7,1,1,1,1,1}, rank=1≡1 (mod 7)
    {6,6}, rank=4≡4 (mod 7)
    {6,5,1}, rank=3≡3 (mod 7)
    {6,4,2}, rank=3≡3 (mod 7)
    {6,4,1,1}, rank=2≡2 (mod 7)
    {6,3,3}, rank=3≡3 (mod 7)
    {6,3,2,1}, rank=2≡2 (mod 7)
    {6,3,1,1,1}, rank=1≡1 (mod 7)
    {6,2,2,2}, rank=2≡2 (mod 7)
    {6,2,2,1,1}, rank=1≡1 (mod 7)
    {6,2,1,1,1,1}, rank=0≡0 (mod 7)
    {6,1,1,1,1,1,1}, rank=-1≡6 (mod 7)
    {5,5,2}, rank=2≡2 (mod 7)
    {5,5,1,1}, rank=1≡1 (mod 7)
    {5,4,3}, rank=2≡2 (mod 7)
    {5,4,2,1}, rank=1≡1 (mod 7)
    {5,4,1,1,1}, rank=0≡0 (mod 7)
    {5,3,3,1}, rank=1≡1 (mod 7)
    {5,3,2,2}, rank=1≡1 (mod 7)
    {5,3,2,1,1}, rank=0≡0 (mod 7)
    {5,3,1,1,1,1}, rank=-1≡6 (mod 7)
    {5,2,2,2,1}, rank=0≡0 (mod 7)
    {5,2,2,1,1,1}, rank=-1≡6 (mod 7)
    {5,2,1,1,1,1,1}, rank=-2≡5 (mod 7)
    {5,1,1,1,1,1,1,1}, rank=-3≡4 (mod 7)
    {4,4,4}, rank=1≡1 (mod 7)
    {4,4,3,1}, rank=0≡0 (mod 7)
    {4,4,2,2}, rank=0≡0 (mod 7)
    {4,4,2,1,1}, rank=-1≡6 (mod 7)
    {4,4,1,1,1,1}, rank=-2≡5 (mod 7)
    {4,3,3,2}, rank=0≡0 (mod 7)
    {4,3,3,1,1}, rank=-1≡6 (mod 7)
    {4,3,2,2,1}, rank=-1≡6 (mod 7)
    {4,3,2,1,1,1}, rank=-2≡5 (mod 7)
    {4,3,1,1,1,1,1}, rank=-3≡4 (mod 7)
    {4,2,2,2,2}, rank=-1≡6 (mod 7)
    {4,2,2,2,1,1}, rank=-2≡5 (mod 7)
    {4,2,2,1,1,1,1}, rank=-3≡4 (mod 7)
    {4,2,1,1,1,1,1,1}, rank=-4≡3 (mod 7)
    {4,1,1,1,1,1,1,1,1}, rank=-5≡2 (mod 7)
    {3,3,3,3}, rank=-1≡6 (mod 7)
    {3,3,3,2,1}, rank=-2≡5 (mod 7)
    {3,3,3,1,1,1}, rank=-3≡4 (mod 7)
    {3,3,2,2,2}, rank=-2≡5 (mod 7)
    {3,3,2,2,1,1}, rank=-3≡4 (mod 7)
    {3,3,2,1,1,1,1}, rank=-4≡3 (mod 7)
    {3,3,1,1,1,1,1,1}, rank=-5≡2 (mod 7)
    {3,2,2,2,2,1}, rank=-3≡4 (mod 7)
    {3,2,2,2,1,1,1}, rank=-4≡3 (mod 7)
    {3,2,2,1,1,1,1,1}, rank=-5≡2 (mod 7)
    {3,2,1,1,1,1,1,1,1}, rank=-6≡1 (mod 7)
    {3,1,1,1,1,1,1,1,1,1}, rank=-7≡0 (mod 7)
    {2,2,2,2,2,2}, rank=-4≡3 (mod 7)
    {2,2,2,2,2,1,1}, rank=-5≡2 (mod 7)
    {2,2,2,2,1,1,1,1}, rank=-6≡1 (mod 7)
    {2,2,2,1,1,1,1,1,1}, rank=-7≡0 (mod 7)
    {2,2,1,1,1,1,1,1,1,1}, rank=-8≡6 (mod 7)
    {2,1,1,1,1,1,1,1,1,1,1}, rank=-9≡5 (mod 7)
    {1,1,1,1,1,1,1,1,1,1,1,1}, rank=-11≡3 (mod 7)

 


재미있는 사실

  • In 1944, the crank was first hinted at by Freeman Dyson (2), then an undergraduate at Cambridge University. He had written an article, titled Some Guesses in the Theory of Partitions, for Eureka, the undergraduate mathematics journal of Cambridge.


 

 

메모

 

 

관련된 항목들

 

 

수학용어번역

 


 

관련논문

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