A330372 Irregular triangle read by rows in which row n lists the self-conjugate partitions of n, ordered by their k-th largest parts, or 0 if such partitions does not exist.

0, 1, 0, 2, 1, 2, 2, 3, 1, 1, 3, 2, 1, 4, 1, 1, 1, 4, 2, 1, 1, 3, 3, 2, 5, 1, 1, 1, 1, 3, 3, 3, 5, 2, 1, 1, 1, 4, 3, 2, 1, 6, 1, 1, 1, 1, 1, 4, 3, 3, 1, 6, 2, 1, 1, 1, 1, 5, 3, 2, 1, 1, 4, 4, 2, 2, 7, 1, 1, 1, 1, 1, 1, 5, 3, 3, 1, 1, 4, 4, 3, 2

Offset: 0

Author

Omar E. Pol, Dec 17 2019

Keywords

• nonn
• tabf

Comments

Row n lists the partitions of n whose Ferrers diagrams are symmetrics.

The k-th part of a partition equals the number of parts >= k of its conjugate partition. Hence, the k-th part of a self-conjugate partition equals the number of parts >= k.

The k-th rank of a partition is the k-th part minus the number of parts >= k. Thus all ranks of a conjugate-partitions are zero. Therefore row n lists the partitions of n whose n ranks are zero, n >= 1. For more information about the k-th ranks see A208478.

Examples

			Triangle begins (rows n = 0..10):
[0];
[1];
[0];
[2, 1];
[2, 2];
[3, 1, 1];
[3, 2, 1];
[4, 1, 1, 1];
[4, 2, 1, 1], [3, 3, 2];
[5, 1, 1, 1, 1], [3, 3, 3];
[5, 2, 1, 1, 1], [4, 3, 2, 1];
...
For n = 10 there are only two partitions of 10 whose Ferrers diagram are symmetric, they are [5, 2, 1, 1, 1] and [4, 3, 2, 1] as shown below:
  * * * * *
  * *
  *
  *
  *
            * * * *
            * * *
            * *
            *
So these partitions form the 10th row of triangle.
On the other hand, only two partitions of 10 have all their ranks equal to zero, they are [5, 2, 1, 1, 1] and [4, 3, 2, 1], so these partitions form the 10th row of triangle.
		

Links

• Freddy Barrera, Rows n = 0..50, flattened

Crossrefs

Row n contains A000700(n) partitions.

The number of positive terms in row n is A067619(n).

Row sums give A330373.

Column 2 gives A000034.

Column 3 gives A000012.

For "k-th rank" of a partition see also: A181187, A208478, A208479, A208482, A208483, A330370.

Extensions

More terms from Freddy Barrera, Dec 31 2019