A330373 -id:A330373 - cp's OEIS frontend

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.

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.

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

A235324 Sum of all parts of all partitions of n into an even number of parts minus the sum of all parts of all partitions of n into an odd number of parts.

Original entry on oeis.org

-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

Offset: 1

Clark Kimberling, Mar 27 2014

Zero together with the absolute values of this sequence give A330373. - Omar E. Pol, Dec 31 2019

			There are 12 partitions of 8 into an even number of parts and 10 into an odd number of parts, so that a(8) = 12*8 - 10*8 = 16.

Clark Kimberling, Table of n, a(n) for n = 1..500

Cf. A000041, A027187, A027193, A081362, A330373.

Table[n*SeriesCoefficient[Product[1-x^k, {k, 1, n, 2}], {x,0,n}], {n,70}]
nmax = 60; Rest[CoefficientList[Series[x*D[Product[1/(1 + x^k), {k, 1, nmax}], x], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 21 2016 *)

a(n)= my(A); n*if(n<0, 0, A= x*O(x^n); polcoeff(eta(x+A)/eta(x^2+A), n));
vector(50, n, a(n) ) /* from Michael Somos code in A081362 */ \\ G. C. Greubel, Dec 31 2019

a(n) = n*A081362(n) for n >= 1.
G.f.: x*f'(x), where f(x) = Product_{k > 0} (1 - x^(2*k - 1)).
a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) * n^(1/4) / (2^(7/4) * 3^(1/4)). - Vaclav Kotesovec, Nov 21 2016

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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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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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A235324 Sum of all parts of all partitions of n into an even number of parts minus the sum of all parts of all partitions of n into an odd number of parts.

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