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.

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, 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, 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

Offset: 1

Author

Omar E. Pol, Dec 18 2019

Keywords

• nonn
• tabl

Comments

The rank of a partition is the largest part minus the number of parts.

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.

Examples

			Triangle begins:
--------------------------------------------------------------------
  n \ k   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14
--------------------------------------------------------------------
[ 1]      1;
[ 2]      0,  2;
[ 3]      1,  0,  2;
[ 4]      1,  2,  0,  2;
[ 5]      1,  2,  2,  0,  2;
[ 6]      1,  4,  2,  2,  0,  2;
[ 7]      3,  2,  4,  2,  2,  0,  2;
[ 8]      2,  6,  4,  4,  2,  2,  0,  2;
[ 9]      4,  6,  6,  4,  4,  2,  2,  0,  2;
[10]      4, 10,  6,  8,  4,  4,  2,  2,  0,  2;
[11]      6, 10, 12,  6,  8,  4,  4,  2,  2,  0,  2;
[12]      7, 16, 12, 12,  8,  8,  4,  4,  2,  2,  0,  2;
[13]     11, 16, 18, 14, 12,  8,  8,  4,  4,  2,  2,  0,  2;
[14]     11, 26, 20, 20, 14, 14,  8,  8,  4,  4,  2,  2,  0,  2;
[15]     16, 28, 30, 22, 22, 14, 14,  8,  8,  4,  4,  2,  2,  0,  2;
...
		

Crossrefs

Row sums give A000041, n >= 1.

Leading diagonal gives A040000.

Second diagonal gives A000004.

Column k=0 is A047993.

Cf. A063995, A105805, A194547, A194549, A195822, A208478, A209616, A330368.

Formula

T(n,k) = A063995(n,k)*A040000(k), 0 <= k < n.