A208478 Triangle read by rows: T(n,k) = number of partitions of n with positive k-th rank.

0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 2, 1, 5, 2, 4, 4, 2, 1, 6, 3, 5, 6, 4, 2, 1, 10, 5, 7, 9, 7, 4, 2, 1, 13, 7, 9, 11, 11, 7, 4, 2, 1, 19, 11, 12, 15, 16, 12, 7, 4, 2, 1, 25, 16, 15, 19, 22, 18, 12, 7, 4, 2, 1, 35, 24, 20, 26, 29, 27, 19, 12, 7, 4, 2, 1

Offset: 1

Author

Omar E. Pol, Mar 07 2012

Keywords

• nonn
• tabl

Comments

We define the k-th rank of a partition as the k-th part minus the number of parts >= k. Every partition of n has n ranks. This is a generalization of the Dyson's rank of a partition which is the largest part minus the number of parts. Since the first part of a partition is also the largest part of the same partition so the Dyson's rank of a partition is the case for k = 1.

The sum of the k-th ranks of all partitions of n is equal to zero.

Also T(n,k) = number of partitions of n with negative k-th rank.

It appears that reversed rows converge to A000070, the same as A208482. - Omar E. Pol, Mar 11 2012

From Omar E. Pol, Dec 12 2019: (Start)

1) The k-th part of a partition of n is also the number of parts >= k of its conjugate partition.

2) The k-th rank of a partitions is also the number of parts >= k of its conjugate partition minus the number of parts >= k.

For example: for n = 9 consider the partition [5, 3, 1]. The first part is 5, so the conjugate partition has five parts >= 1. The second part is 3, so the conjugate partition has three parts >= 2. The third part is 1, so the conjugate partition has only one part >= 3. The mentioned conjugate partition is [3, 2, 2, 1, 1]. And conversely, consider the partition [3, 2, 2, 1, 1]. The first part is 3, so the conjugate partition has three parts >= 1. The second part is 2, so the conjugate partition has two parts >= 2. the Third part is 2, so the conjugate partition has two parts >= 3, and so on. In this case the conjugate partition is [5, 3, 1].

3) The difference between the k-th part and the (k+1)-st part of the partition of n is also the number of k's in its conjugate partition. For example: consider the partition [5, 3, 1]. The difference between the first and the second part is 5 - 3 = 2, equals the number of 1's in its conjugate partition. The difference between the second and the third part is 3 - 1 = 2, equals the number of 2's in its conjugate partition. The difference between the third and the fourth (virtual) part is 1 - 0 = 1, equals the number of 3's in its conjugate partition [3, 2, 2, 1, 1]. And conversely, consider the partition [3, 2, 2, 1, 1]. The difference between the first and the second part is 3 - 2 = 1, equals the number of 1's in its conjugate partition. The difference between the second and the third part is 2 - 2 = 0, equals the number of 2's in its conjugate partition. The difference between the third and the fourth part is 2 - 1 = 1, equals the number of 3's in its conjugate partition, and so on.

4) The list of n ranks of a partition of n equals the list of n ranks multiplied by -1 of its conjugate partition. For example the nine ranks of the partition [5, 3, 1] of 9 are [2, 1, -1, -1, -1, -1, 0, 0, 0], and the nine ranks of its conjugate partition [3, 2, 2, 1, 1] are [-2, -1, 1, 1, 1, 1, 0, 0, 0].

For a list of partitions of the positive integers ordered by its k-th ranks see A330370. (End)

Examples

			For n = 4 the partitions of 4 and the four types of ranks of the partitions of 4 are
----------------------------------------------------------
Partitions    First      Second       Third      Fourth
of 4          rank        rank        rank        rank
----------------------------------------------------------
4           4-1 =  3    0-1 = -1    0-1 = -1    0-1 = -1
3+1         3-2 =  1    1-1 =  0    0-1 = -1    0-0 =  0
2+2         2-2 =  0    2-2 =  0    0-0 =  0    0-0 =  0
2+1+1       2-3 = -1    1-1 =  0    1-0 =  1    0-0 =  0
1+1+1+1     1-4 = -3    1-0 =  1    1-0 =  1    1-0 =  1
----------------------------------------------------------
The number of partitions of 4 with positive k-th ranks are 2, 1, 2, 1 so row 4 lists 2, 1, 2, 1.
Triangle begins:
   0;
   1,  1;
   1,  1,  1;
   2,  1,  2,  1;
   3,  1,  3,  2,  1;
   5,  2,  4,  4,  2,  1;
   6,  3,  5,  6,  4,  2,  1;
  10,  5,  7,  9,  7,  4,  2,  1;
  13,  7,  9, 11, 11,  7,  4,  2,  1;
  19, 11, 12, 15, 16, 12,  7,  4,  2,  1;
  25, 16, 15, 19, 22, 18, 12,  7,  4,  2,  1;
  35, 24, 20, 26, 29, 27, 19, 12,  7,  4,  2,  1;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..44, flattened

Crossrefs

Column 1 is A064173.

Row sums give A208479.

Cf. A063995, A105805, A181187, A194547, A194549, A195822, A208482, A208483, A209616, A330368, A330369, A330370.

Extensions

More terms from Alois P. Heinz, Mar 11 2012