

A208478


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


12



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
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OFFSET

1,7


COMMENTS

We define the kth rank of a partition as the kth 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 kth ranks of all partitions of n is equal to zero.
Also T(n,k) = number of partitions of n with negative kth 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 kth part of a partition of n is also the number of parts >= k of its conjugate partition.
2) The kth 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 kth 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 kth ranks see A330370. (End)


LINKS

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


EXAMPLE

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 41 = 3 01 = 1 01 = 1 01 = 1
3+1 32 = 1 11 = 0 01 = 1 00 = 0
2+2 22 = 0 22 = 0 00 = 0 00 = 0
2+1+1 23 = 1 11 = 0 10 = 1 00 = 0
1+1+1+1 14 = 3 10 = 1 10 = 1 10 = 1

The number of partitions of 4 with positive kth 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;
...


CROSSREFS

Column 1 is A064173.
Row sums give A208479.
Cf. A063995, A105805, A181187, A194547, A194549, A195822, A208482, A208483, A209616, A330368, A330369, A330370.
Sequence in context: A027352 A029238 A345137 * A274450 A126131 A343332
Adjacent sequences: A208475 A208476 A208477 * A208479 A208480 A208481


KEYWORD

nonn,tabl


AUTHOR

Omar E. Pol, Mar 07 2012


EXTENSIONS

More terms from Alois P. Heinz, Mar 11 2012


STATUS

approved



