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

Author

Omar E. Pol, Dec 12 2019

Keywords

• nonn
• tabf

Comments

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).

Examples

			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.
		

Crossrefs

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.