"자연수의 분할(partition)과 rank/crank 목록"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
Pythagoras0 (토론 | 기여) |
|||
| (사용자 2명의 중간 판 33개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
| − | + | ==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 | |
| + | * <nowiki>{{1}}</nowiki> | ||
| − | + | ||
| − | + | ==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. | ||
| + | |||
| + | |||
| + | * [[수학사 연표]] | ||
| + | |||
| + | |||
| + | |||
| + | |||
| − | + | ==메모== | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | ||
| − | + | ||
| − | + | ==관련된 항목들== | |
| − | |||
| − | + | ||
| − | + | ||
| − | + | ==수학용어번역== | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | + | * http://www.google.com/dictionary?langpair=en|ko&q= | |
| + | * [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집] | ||
| + | ** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr= | ||
| + | * [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판] | ||
| − | + | ||
| − | |||
| − | + | ||
| − | |||
| − | |||
| − | |||
| − | + | ==관련논문== | |
| + | * 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] | ||
| + | ** 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] | ||
| + | ** George E. Andrews and F. G. Garvan, Bull. Amer. Math. Soc. (N.S.) Volume 18, Number 2 (1988), 167-171 | ||
| − | |||
| − | |||
| − | + | [[분류:q-급수]] | |
| − | + | [[분류:목록]] | |
| − | + | [[분류:분할수]] | |
| − | |||
| − | |||
| − | |||
2020년 12월 28일 (월) 02:53 기준 최신판
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
- Dousse, Jehanne, and Michael H. Mertens. “Asymptotic Formulae for Partition Ranks.” arXiv:1406.6848 [math], June 26, 2014. http://arxiv.org/abs/1406.6848.
- Ramanujan's congruences and Dyson's crank
- George E. Andrews and Ken Ono, PNAS October 25, 2005 vol. 102 no. 43 15277
- Dyson's crank of a partition
- George E. Andrews and F. G. Garvan, Bull. Amer. Math. Soc. (N.S.) Volume 18, Number 2 (1988), 167-171