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A330374 Triangle read by rows: T(n,k) is the number of partitions of n whose absolute value of Dyson's rank is equal to k, with 0 <= k < n.

%I A330374 #24 Dec 31 2019 08:26:35
%S A330374 1,0,2,1,0,2,1,2,0,2,1,2,2,0,2,1,4,2,2,0,2,3,2,4,2,2,0,2,2,6,4,4,2,2,
%T A330374 0,2,4,6,6,4,4,2,2,0,2,4,10,6,8,4,4,2,2,0,2,6,10,12,6,8,4,4,2,2,0,2,7,
%U A330374 16,12,12,8,8,4,4,2,2,0,2,11,16,18,14,12,8,8,4,4,2,2,0,2,11,26,20,20,14,14
%N A330374 Triangle read by rows: T(n,k) is the number of partitions of n whose absolute value of Dyson's rank is equal to k, with 0 <= k < n.
%C A330374 The rank of a partition is the largest part minus the number of parts.
%C A330374 Since the largest part of a partition equals the number of parts of its conjugate partition, so the rank of a partition also is equal to the difference between the number of parts of its conjugate partition and the number of parts of the partition.
%F A330374 T(n,k) = A063995(n,k)*A040000(k), 0 <= k < n.
%e A330374 Triangle begins:
%e A330374 --------------------------------------------------------------------
%e A330374   n \ k   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14
%e A330374 --------------------------------------------------------------------
%e A330374 [ 1]      1;
%e A330374 [ 2]      0,  2;
%e A330374 [ 3]      1,  0,  2;
%e A330374 [ 4]      1,  2,  0,  2;
%e A330374 [ 5]      1,  2,  2,  0,  2;
%e A330374 [ 6]      1,  4,  2,  2,  0,  2;
%e A330374 [ 7]      3,  2,  4,  2,  2,  0,  2;
%e A330374 [ 8]      2,  6,  4,  4,  2,  2,  0,  2;
%e A330374 [ 9]      4,  6,  6,  4,  4,  2,  2,  0,  2;
%e A330374 [10]      4, 10,  6,  8,  4,  4,  2,  2,  0,  2;
%e A330374 [11]      6, 10, 12,  6,  8,  4,  4,  2,  2,  0,  2;
%e A330374 [12]      7, 16, 12, 12,  8,  8,  4,  4,  2,  2,  0,  2;
%e A330374 [13]     11, 16, 18, 14, 12,  8,  8,  4,  4,  2,  2,  0,  2;
%e A330374 [14]     11, 26, 20, 20, 14, 14,  8,  8,  4,  4,  2,  2,  0,  2;
%e A330374 [15]     16, 28, 30, 22, 22, 14, 14,  8,  8,  4,  4,  2,  2,  0,  2;
%e A330374 ...
%Y A330374 Row sums give A000041, n >= 1.
%Y A330374 Leading diagonal gives A040000.
%Y A330374 Second diagonal gives A000004.
%Y A330374 Column k=0 is A047993.
%Y A330374 Cf. A063995, A105805, A194547, A194549, A195822, A208478, A209616, A330368.
%K A330374 nonn,tabl
%O A330374 1,3
%A A330374 _Omar E. Pol_, Dec 18 2019