A299775 Irregular triangle read by rows in which row n lists the indices of the partitions into consecutive parts in the list of colexicographically ordered partitions of n.

1, 2, 2, 3, 5, 6, 7, 6, 11, 14, 15, 22, 25, 29, 30, 25, 42, 55, 56

Offset: 1

Author

Omar E. Pol, Mar 29 2018

Keywords

• nonn
• more
• tabf

Comments

If n > 1 and n is odd then row n ending in [p(n) - 1, p(n)], where p(n) is A000041(n).

Examples

			Triangle begins:
   1;
   2;
   2,  3;
   5;
   6,  7;
   6, 11;
  14, 15;
  22;
  25, 29, 30;
  25, 42;
  55, 56;
...
For n = 9 the partitions of 9 into consecutive parts are [4, 3, 2], [5, 4] and [9]. Then we have that in the list of colexicographically ordered partitions of 9 these partitions are in the rows 25, 29 and 30 respectively as shown below, so the 9th row of the triangle is [25, 29, 30].
--------------------------------------------------------
   p         Diagram          Partitions of 9
--------------------------------------------------------
        1 2 3 4 5 6 7 8 9
        _ _ _ _ _ _ _ _ _
   1   |_| | | | | | | | |   [1, 1, 1, 1, 1, 1, 1, 1, 1]
   2   |_ _| | | | | | | |   [2, 1, 1, 1, 1, 1, 1, 1]
   3   |_ _ _| | | | | | |   [3, 1, 1, 1, 1, 1, 1]
   4   |_ _|   | | | | | |   [2, 2, 1, 1, 1, 1, 1]
   5   |_ _ _ _| | | | | |   [4, 1, 1, 1, 1, 1]
   6   |_ _ _|   | | | | |   [3, 2, 1, 1, 1, 1]
   7   |_ _ _ _ _| | | | |   [5, 1, 1, 1, 1]
   8   |_ _|   |   | | | |   [2, 2, 2, 1, 1, 1]
   9   |_ _ _ _|   | | | |   [4, 2, 1, 1, 1]
  10   |_ _ _|     | | | |   [3, 3, 1, 1, 1]
  11   |_ _ _ _ _ _| | | |   [6, 1, 1, 1]
  12   |_ _ _|   |   | | |   [3, 2, 2, 1, 1]
  13   |_ _ _ _ _|   | | |   [5, 2, 1, 1]
  14   |_ _ _ _|     | | |   [4, 3, 1, 1]
  15   |_ _ _ _ _ _ _| | |   [7, 1, 1]
  16   |_ _|   |   |   | |   [2, 2, 2, 2, 1]
  17   |_ _ _ _|   |   | |   [4, 2, 2, 1]
  18   |_ _ _|     |   | |   [3, 3, 2, 1]
  19   |_ _ _ _ _ _|   | |   [6, 2, 1]
  20   |_ _ _ _ _|     | |   [5, 3, 1]
  21   |_ _ _ _|       | |   [4, 4, 1]
  22   |_ _ _ _ _ _ _ _| |   [8, 1]
  23   |_ _ _|   |   |   |   [3, 2, 2, 2]
  24   |_ _ _ _ _|   |   |   [5, 2, 2]
  25   |_ _ _ _|     |   |   [4, 3, 2]   <--- Consecutive parts
  26   |_ _ _ _ _ _ _|   |   [7, 2]
  27   |_ _ _|     |     |   [3, 3, 3]
  28   |_ _ _ _ _ _|     |   [6, 3]
  29   |_ _ _ _ _|       |   [5, 4]   <--- Consecutive parts
  30   |_ _ _ _ _ _ _ _ _|   [9]   <--- Consecutive parts
.
		

Crossrefs

Row n has length A001227(n).

Right border gives A000041, n >= 1.

Cf. A211992 (partitions in colexicographic order).

Cf. A299765 (partitions into consecutive parts).

For tables of partitions into consecutive parts see also A286000 and A286001.

Cf. A000041, A135010, A141285, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A299474, A299475, A299773, A299774.