OFFSET
1,7
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.
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)
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 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;
...
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Mar 07 2012
EXTENSIONS
More terms from Alois P. Heinz, Mar 11 2012
STATUS
approved