자연수의 분할(partition)과 rank/crank 목록

수학노트
이동: 둘러보기, 검색

개요

  • 분할의 rank = 분할에서 가장 큰 수 - 분할의 크기

    • 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)

    • 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.


 

 

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