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.

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

Author

Omar E. Pol, Dec 15 2019

Keywords

• nonn
• tabf

Comments

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).

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], [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
		

Crossrefs

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.