예
분할의 crank
예
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
{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)
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}
{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)
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)
네이버 지식인
The On-Line Encyclopedia of Integer Sequences
Ramanujan's congruences and Dyson's crank
도서내검색
도서검색
네이버 뉴스 검색 (키워드 수정)
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