A208479 -id:A208479 - cp's OEIS frontend

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

Omar E. Pol, Mar 07 2012

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)

			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;
  ...

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

Column 1 is A064173.
Row sums give A208479.
Cf. A063995, A105805, A181187, A194547, A194549, A195822, A208482, A208483, A209616, A330368, A330369, A330370.

More terms from Alois P. Heinz, Mar 11 2012

A208483 Total sum of the sums of all positive k-th ranks of all partitions of n.

Original entry on oeis.org

0, 2, 4, 8, 14, 26, 40, 68, 100, 156, 224, 334, 466, 668, 920, 1278, 1726, 2356, 3130, 4190, 5508, 7254, 9422, 12268, 15764, 20284, 25852, 32934, 41616, 52578, 65938, 82648, 102976, 128144, 158660, 196222, 241534, 296946, 363632, 444650, 541794, 659268, 799606

Offset: 1

Omar E. Pol, Mar 07 2012

nonn

For the definition of k-th rank see A208478.

			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 sums of positive k-th ranks of the partitions of 4 are 4, 1, 2, 1 so the total sum is a(4) = 4+1+2+1 = 8.

Row sums of triangle A208482.

Cf. A208478, A208479.

More terms from Alois P. Heinz, Mar 11 2012

A330370 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = 1..n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 12 2019

Theorem: the k-th part of a partition in nonincreasing order of a positive integer equals the number of parts >= k of its conjugate partition.
Example: for n = 9 consider the partition [5, 3, 1]. The first part is 5, so the conjugate partition [3, 2, 2, 1, 1] 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. And vice versa, consider now the partition [3, 2, 2, 1, 1]. The first part is 3, so the conjugate partition [5, 3, 1] 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. The fourth part is 1, so the conjugate partition has only one part >= 4. The fifth part is 1, so the conjugate partition has only one part >= 5.
Corollary: the difference between the k-th part and the (k+1)-st part of a partition in nonincreasing order of a positive integer equals the number of k's in its conjugate partition.
Example: consider the partition [5, 3, 1]. The difference between the first and the second parts is 5 - 3 = 2, which equals the number of 1's in its conjugate partition [3, 2, 2, 1, 1]. The difference between the second and third parts is 3 - 1 = 2, which equals the number of 2's in its conjugate partition. The difference between the third part and the fourth (virtual) part is 1 - 0 = 1, which equals the number of 3's in its conjugate partition. And vice versa, consider the partition [3, 2, 2, 1, 1]. The difference between the first and second parts is 3 - 2 = 1, which equals the number of 1's in its conjugate partition [5, 3, 1]. The difference between the second and third parts is 2 - 2 = 0, which equals the number of 2's in its conjugate partition. The difference between the third and fourth parts is 2 - 1 = 1, which equals the number of 3's in its conjugate partition, and so on.
Self-conjugate partitions are included in all the above comments.
A proof without words is as shown below:
.
                       +------------------------+
                       | +--------------------+ |
                       | | +----------------+ | |
                       | | |                | | |
                       v v v   P2  FD  k    | | |
                                            | | |
           +-------->  * * *   3   1   1  --+ | |
           | +------>  * *     2   0   2      | |
           | +------>  * *     2   1   3  ----+ |
           | | +---->  *       1   0   4        |
           | | +---->  *       1   1   5  ------+
           | | |
           | | |  P1   5 3 1
           | | |
           | | |  FD   2 2 1
           | | |
           | | |  k    1 2 3
           | | |
           | | |       | | |
           | | +-------+ | |
           | +-----------+ |
           +---------------+
.
Every partition of n has n ranks.
The k-th rank of a partition is the k-th part minus the number of parts >= k.
In accordance with the above theorem, the k-th rank of a partition is also the number of parts >= k of its conjugate partition minus the number of parts >= k of the partition.
All ranks of a partition are zeros if and only if the partition is a self-conjugate partition.
The list of ranks of a partition of n equals the list of ranks multipled by -1 of its conjugate partition.
For example, the nine ranks of the partition [5, 3, 1] 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].
Note that the first rank coincides with the Dyson's rank because the first part of a partition is also the largest part, and the number of parts >= 1 is also the total number of parts.
In this triangle the partitions of n appears ordered by their first rank. The partitions that have the same first rank appears ordered by their second rank. The partitions that have the same first rank and the same second rank appears ordered by their third rank, and so on. The partitions that have all k-ranks equal zero appears ordered by their largest parts, then by their second largest parts, then by their third largest parts, and so on.
Note that a partition and its conjugate partition both are equidistants from the center of the list of partitions of n.
The first ranks of the partitions of this triangle give A330368.
For more information about the k-th ranks see A208478.
First differs from A080577 at a(48), and from A036037 at a(56), and from A181317 at a(105).

			Triangle begins:
  [1];
  [2], [1,1];
  [3], [2,1], [1,1,1];
  [4], [3,1], [2,2], [2,1,1], [1,1,1,1];
  [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1];
  [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [3,1,1,1], [2,2,2], ...
  ...
Illustration of initial terms with a symmetric arrangement (note that the self-conjugate partitions are located in the main diagonal):
.
  1    1 1    1 1 1    1 1 1 1    1 1 1 1 1           1 1 1 1 1 1
  *    * *    * * *    * * * *    * * * * *           * * * * * *
  2
  *
  *
  3           2 1      2 1 1      2 1 1 1             2 1 1 1 1
  *           * *      * * *      * * * *             * * * * *
  *           *        *          *                   *
  *
  4           3 1      2 2        2 2 1               2 2 1 1
  *           * *      * *        * * *               * * * *
  *           *        * *        * *                 * *
  *           *
  *
  5           4 1      3 2        3 1 1               2 2 2
  *           * *      * *        * * *               * * *
  *           *        * *        *                   * * *
  *           *        *          *
  *           *                                       3 1 1 1
  *                                                   * * * *
                                                      *
                                                      *
.
  6           5 1      4 2        3 3      4 1 1      3 2 1
  *           * *      * *        * *      * * *      * * *
  *           *        * *        * *      *          * *
  *           *        *          * *      *          *
  *           *        *                   *
  *           *
  *
For n = 9 the 9th row of the triangle contains the partitions ordered as shown below:
---------------------------------------------------------------------------------
                                                                Ranks
          Conjugate
Label       with        Partitions                k = 1  2  3  4  5  6  7  8  9
---------------------------------------------------------------------------------
   1         30         [9]                           8 -1 -1 -1 -1 -1 -1 -1 -1
   2         29         [8, 1]                        6  0 -1 -1 -1 -1 -1 -1  0
   3         28         [7, 2]                        5  0 -1 -1 -1 -1 -1  0  0
   4         27         [6, 3]                        4  1 -2 -1 -1 -1  0  0  0
   5         26         [7, 1, 1]                     4  0  0 -1 -1 -1 -1  0  0
   6         25         [5, 4]                        3  2 -2 -2 -1  0  0  0  0
   7         24         [6, 2, 1]                     3  0  0 -1 -1 -1  0  0  0
   8         23         [5, 3, 1]                     2  1 -1 -1 -1  0  0  0  0
   9         22         [6, 1, 1, 1]                  2  0  0  0 -1 -1  0  0  0
  10         21         [5, 2, 2]                     2 -1  1 -1 -1  0  0  0  0
  11         20         [4, 4, 1]                     1  2 -1 -2  0  0  0  0  0
  12         19         [5, 2, 1, 1]                  1  0  0  0 -1  0  0  0  0
  13         18         [4, 3, 2]                     1  0  0 -1  0  0  0  0  0
  14         17         [4, 3, 1, 1]                  0  1 -1  0  0  0  0  0  0
  15  (self-conjugate)  [5, 1, 1, 1, 1]  All zeros -> 0  0  0  0  0  0  0  0  0
  16  (self-conjugate)  [3, 3, 3]        All zeros -> 0  0  0  0  0  0  0  0  0
  17         14         [4, 2, 2, 1]                  0 -1  1  0  0  0  0  0  0
  18         13         [3, 3, 2, 1]                 -1  0  0  1  0  0  0  0  0
  19         12         [4, 2, 1, 1, 1]              -1  0  0  0  1  0  0  0  0
  20         11         [3, 2, 2, 2]                 -1 -2  1  2  0  0  0  0  0
  21         10         [3, 3, 1, 1, 1]              -2  1 -1  1  1  0  0  0  0
  22          9         [4, 1, 1, 1, 1, 1]           -2  0  0  0  1  1  0  0  0
  23          8         [3, 2, 2, 1, 1]              -2 -1  1  1  1  0  0  0  0
  24          7         [3, 2, 1, 1, 1, 1]           -3  0  0  1  1  1  0  0  0
  25          6         [2, 2, 2, 2, 1]              -3 -2  2  2  1  0  0  0  0
  26          5         [3, 1, 1, 1, 1, 1, 1]        -4  0  0  1  1  1  1  0  0
  27          4         [2, 2, 2, 1, 1, 1]           -4 -1  2  1  1  1  0  0  0
  28          3         [2, 2, 1, 1, 1, 1, 1]        -5  0  1  1  1  1  1  0  0
  29          2         [2, 1, 1, 1, 1, 1, 1, 1]     -6  0  1  1  1  1  1  1  0
  30          1         [1, 1, 1, 1, 1, 1, 1, 1, 1]  -8  1  1  1  1  1  1  1  1
.
Two examples of the order of partitions:
1) The partitions [6, 3] and [7, 1, 1] both have their first rank equal to 4, so they are ordered by their sencond rank.
2) The self-conjugate partitions [5, 1, 1, 1, 1] and [3, 3, 3] both have all their ranks equal to zero, so they are ordered by their first part.

Row n contains A000041(n) partitions.
Row n has length A006128(n).
The sum of n-th row is A066186(n).
Cf. A000700, A036037, A067619, A080577, A181317, A207031, A330368, A330372, A330373.
For "k-th rank" of a partition see also: A181187, A208478, A208479, A208482, A208483.

A330373 Sum of all parts of all self-conjugate partitions of n.

Original entry on oeis.org

0, 1, 0, 3, 4, 5, 6, 7, 16, 18, 20, 22, 36, 39, 42, 60, 80, 85, 90, 114, 140, 168, 176, 207, 264, 300, 312, 378, 448, 493, 540, 620, 736, 825, 884, 1015, 1188, 1295, 1406, 1599, 1840, 2009, 2184, 2451, 2772, 3060, 3312, 3666, 4176, 4557, 4900, 5457, 6084, 6625, 7182, 7920, 8792, 9576, 10324, 11328, 12540

Offset: 0

Omar E. Pol, Dec 17 2019

nonn

a(n) is the sum of all parts of all partitions of n whose Ferrers diagrams are symmetric.
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, a(n) is also the sum of all parts of all partitions of n whose n ranks are zero, n >= 1. For more information about the k-th ranks see A208478.

			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:
  * * * * *
  * *
  *
  *
  *
            * * * *
            * * *
            * *
            *
The sum of all parts of these partitions is 5 + 2 + 1 + 1 + 1 + 4 + 3 + 2 + 1 = 20, so a(10) = 20.
Also, in accordance with the first formula; a(10) = 2*10 = 20.

Freddy Barrera, Table of n, a(n) for n = 0..10000

Row sums of A330372.
For "k-th rank" of a partition see also: A181187, A208478, A208479, A208482, A208483, A330370.
Cf. A000700, A066186, A081362, A235324.

seq(n)={Vec(deriv(exp(sum(k=1, n, x^k/(k*(1 - (-x)^k)) + O(x*x^n)))), -(n+1))} \\ Andrew Howroyd, Dec 31 2019

a(n) = n*A000700(n).
a(n) = abs(n*A081362(n)).
a(n) = abs(A235324(n)), n >= 1.

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.

Original entry on oeis.org

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

Omar E. Pol, Dec 17 2019

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.

			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.

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

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.

More terms from Freddy Barrera, Dec 31 2019

A330371 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by the lower value of their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = n..1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 15 2019

In this triangle the partitions of n are ordered by their n-th rank. The partitions that have the same n-th rank appears ordered by their (n-1)-st rank. The partitions that have the same n-th rank and the same (n-1)-st rank appears ordered by their (n-2)-nd rank, and so on. The partitions that have all k-ranks equal zero appears ordered by their largest parts, then by their second largest parts, then by their third largest parts, and so on.
Note that a partition and its conjugate partition both are equidistants from the center of the list of partitions of n.
For further information see A330370.
First differs from A036037, A181317, A330370 and A334439 at a(48).
First differs from A080577 at a(56).

			Triangle begins:
[1];
[2], [1,1];
[3], [2,1], [1,1,1];
[4], [3,1], [2,2], [2,1,1], [1,1,1,1];
[5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1];
[6], [5,1], [4,2], [4,1,1], [3,3], [3,2,1], [2,2,2], [3,1,1,1], [2,2,1,1], ...
.
For n = 9 the 9th row of the triangle contains the partitions ordered as shown below:
---------------------------------------------------------------------------------
                                                                Ranks
          Conjugate
Label     with label    Partition                 k = 1  2  3  4  5  6  7  8  9
---------------------------------------------------------------------------------
   1         30         [9]                           8 -1 -1 -1 -1 -1 -1 -1 -1
   2         29         [8, 1]                        6  0 -1 -1 -1 -1 -1 -1  0
   3         28         [7, 2]                        5  0 -1 -1 -1 -1 -1  0  0
   4         27         [7, 1, 1]                     4  0  0 -1 -1 -1 -1  0  0
   5         26         [6, 3]                        4  1 -2 -1 -1 -1  0  0  0
   6         25         [6, 2, 1]                     3  0  0 -1 -1 -1  0  0  0
   7         24         [6, 1, 1, 1]                  2  0  0  0 -1 -1  0  0  0
   8         23         [5, 4]                        3  2 -2 -2 -1  0  0  0  0
   9         22         [5, 3, 1]                     2  1 -1 -1 -1  0  0  0  0
  10         21         [5, 2, 2]                     2 -1  1 -1 -1  0  0  0  0
  11         20         [5, 2, 1, 1]                  1  0  0  0 -1  0  0  0  0
  12         19         [4, 4, 1]                     1  2 -1 -2  0  0  0  0  0
  13         18         [4, 3, 2]                     1  0  0 -1  0  0  0  0  0
  14         17         [4, 3, 1, 1]                  0  1 -1  0  0  0  0  0  0
  15  (self-conjugate)  [5, 1, 1, 1, 1]  All zeros -> 0  0  0  0  0  0  0  0  0
  16  (self-conjugate)  [3, 3, 3]        All zeros -> 0  0  0  0  0  0  0  0  0
  17         14         [4, 2, 2, 1]                  0 -1  1  0  0  0  0  0  0
  18         13         [3, 3, 2, 1]                 -1  0  0  1  0  0  0  0  0
  19         12         [3, 2, 2, 2]                 -1 -2  1  2  0  0  0  0  0
  20         11         [4, 2, 1, 1, 1]              -1  0  0  0  1  0  0  0  0
  21         10         [3, 3, 1, 1, 1]              -2  1 -1  1  1  0  0  0  0
  22          9         [3, 2, 2, 1, 1]              -2 -1  1  1  1  0  0  0  0
  23          8         [2, 2, 2, 2, 1]              -3 -2  2  2  1  0  0  0  0
  24          7         [4, 1, 1, 1, 1, 1]           -2  0  0  0  1  1  0  0  0
  25          6         [3, 2, 1, 1, 1, 1]           -3  0  0  1  1  1  0  0  0
  26          5         [2, 2, 2, 1, 1, 1]           -4 -1  2  1  1  1  0  0  0
  27          4         [3, 1, 1, 1, 1, 1, 1]        -4  0  0  1  1  1  1  0  0
  28          3         [2, 2, 1, 1, 1, 1, 1]        -5  0  1  1  1  1  1  0  0
  29          2         [2, 1, 1, 1, 1, 1, 1, 1]     -6  0  1  1  1  1  1  1  0
  30          1         [1, 1, 1, 1, 1, 1, 1, 1, 1]  -8  1  1  1  1  1  1  1  1

Another version of A330370.
Row n contains A000041(n) partitions.
Row n has length A006128(n).
The sum of n-th row is A066186(n).
Cf. A000700, A036037, A080577, A181317, A207031, A330368, A330370, A334439.
For the "k-th rank" see also: A181187, A208478, A208479, A208482, A208483.

cp's OEIS Frontend

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

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A208483 Total sum of the sums of all positive k-th ranks of all partitions of n.

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A330370 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = 1..n.

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A330373 Sum of all parts of all self-conjugate partitions of n.

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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.

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A330371 Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by the lower value of their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = n..1.

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