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Author: Roan James
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Description: RANK MODULO 7 AND 11
Compute Environment: Ubuntu 20.04 (Default)
#Looking at the Rank Modulo 7 for n in range(1,20): g=Partitions(n).cardinality() countbyrank=[0,0,0,0,0,0,0] for P in Partitions(n): countbyrank[(P[0]-len(P))%7]+=1 print (n, countbyrank)
1 [1, 0, 0, 0, 0, 0, 0] 2 [0, 1, 0, 0, 0, 0, 1] 3 [1, 0, 1, 0, 0, 1, 0] 4 [1, 1, 0, 1, 1, 0, 1] 5 [1, 1, 1, 1, 1, 1, 1] 6 [1, 2, 2, 1, 1, 2, 2] 7 [3, 2, 2, 2, 2, 2, 2] 8 [4, 3, 3, 3, 3, 3, 3] 9 [4, 5, 4, 4, 4, 4, 5] 10 [6, 6, 6, 6, 6, 6, 6] 11 [8, 8, 8, 8, 8, 8, 8] 12 [11, 11, 11, 11, 11, 11, 11] 13 [15, 14, 15, 14, 14, 15, 14] 14 [19, 20, 19, 19, 19, 19, 20] 15 [26, 25, 25, 25, 25, 25, 25] 16 [33, 34, 32, 33, 33, 32, 34] 17 [43, 42, 43, 42, 42, 43, 42] 18 [55, 55, 55, 55, 55, 55, 55] 19 [70, 70, 70, 70, 70, 70, 70]
#Looking at the Rank Modulo 11 for n in range(1,20): g=Partitions(n).cardinality() countbyrank=[0,0,0,0,0,0,0,0,0,0,0] for P in Partitions(n): countbyrank[(P[0]-len(P))%11]+=1 print (n, countbyrank)
1 [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] 2 [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1] 3 [1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0] 4 [1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1] 5 [1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1] 6 [1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2] 7 [3, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1] 8 [2, 3, 2, 2, 2, 1, 1, 2, 2, 2, 3] 9 [4, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3] 10 [4, 5, 4, 4, 3, 3, 3, 3, 4, 4, 5] 11 [6, 6, 6, 4, 5, 4, 4, 5, 4, 6, 6] 12 [9, 8, 7, 7, 6, 6, 6, 6, 7, 7, 8] 13 [11, 10, 10, 9, 8, 8, 8, 8, 9, 10, 10] 14 [13, 14, 13, 12, 11, 11, 11, 11, 12, 13, 14] 15 [18, 17, 17, 16, 15, 14, 14, 15, 16, 17, 17] 16 [23, 23, 21, 21, 20, 19, 19, 20, 21, 21, 23] 17 [29, 29, 28, 26, 26, 25, 25, 26, 26, 28, 29] 18 [37, 37, 36, 35, 33, 33, 33, 33, 35, 36, 37] 19 [48, 46, 46, 44, 43, 42, 42, 43, 44, 46, 46]