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

1,2

Comments

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

Links

Table of n, a(n) for n=1..19.

Example

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.

Sequence in context: A357411 A270875 A117752 * A227191 A214045 A136432

Adjacent sequences: A299772 A299773 A299774 * A299776 A299777 A299778

Keyword

nonn,more,tabf

Author

Omar E. Pol, Mar 29 2018

Status

approved