cp's OEIS Frontend

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.

%I A299775 #28 Aug 26 2018 12:32:17
%S A299775 1,2,2,3,5,6,7,6,11,14,15,22,25,29,30,25,42,55,56
%N 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.
%C A299775 If n > 1 and n is odd then row n ending in [p(n) - 1, p(n)], where p(n) is A000041(n).
%e A299775 Triangle begins:
%e A299775    1;
%e A299775    2;
%e A299775    2,  3;
%e A299775    5;
%e A299775    6,  7;
%e A299775    6, 11;
%e A299775   14, 15;
%e A299775   22;
%e A299775   25, 29, 30;
%e A299775   25, 42;
%e A299775   55, 56;
%e A299775 ...
%e A299775 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].
%e A299775 --------------------------------------------------------
%e A299775    p         Diagram          Partitions of 9
%e A299775 --------------------------------------------------------
%e A299775         1 2 3 4 5 6 7 8 9
%e A299775         _ _ _ _ _ _ _ _ _
%e A299775    1   |_| | | | | | | | |   [1, 1, 1, 1, 1, 1, 1, 1, 1]
%e A299775    2   |_ _| | | | | | | |   [2, 1, 1, 1, 1, 1, 1, 1]
%e A299775    3   |_ _ _| | | | | | |   [3, 1, 1, 1, 1, 1, 1]
%e A299775    4   |_ _|   | | | | | |   [2, 2, 1, 1, 1, 1, 1]
%e A299775    5   |_ _ _ _| | | | | |   [4, 1, 1, 1, 1, 1]
%e A299775    6   |_ _ _|   | | | | |   [3, 2, 1, 1, 1, 1]
%e A299775    7   |_ _ _ _ _| | | | |   [5, 1, 1, 1, 1]
%e A299775    8   |_ _|   |   | | | |   [2, 2, 2, 1, 1, 1]
%e A299775    9   |_ _ _ _|   | | | |   [4, 2, 1, 1, 1]
%e A299775   10   |_ _ _|     | | | |   [3, 3, 1, 1, 1]
%e A299775   11   |_ _ _ _ _ _| | | |   [6, 1, 1, 1]
%e A299775   12   |_ _ _|   |   | | |   [3, 2, 2, 1, 1]
%e A299775   13   |_ _ _ _ _|   | | |   [5, 2, 1, 1]
%e A299775   14   |_ _ _ _|     | | |   [4, 3, 1, 1]
%e A299775   15   |_ _ _ _ _ _ _| | |   [7, 1, 1]
%e A299775   16   |_ _|   |   |   | |   [2, 2, 2, 2, 1]
%e A299775   17   |_ _ _ _|   |   | |   [4, 2, 2, 1]
%e A299775   18   |_ _ _|     |   | |   [3, 3, 2, 1]
%e A299775   19   |_ _ _ _ _ _|   | |   [6, 2, 1]
%e A299775   20   |_ _ _ _ _|     | |   [5, 3, 1]
%e A299775   21   |_ _ _ _|       | |   [4, 4, 1]
%e A299775   22   |_ _ _ _ _ _ _ _| |   [8, 1]
%e A299775   23   |_ _ _|   |   |   |   [3, 2, 2, 2]
%e A299775   24   |_ _ _ _ _|   |   |   [5, 2, 2]
%e A299775   25   |_ _ _ _|     |   |   [4, 3, 2]   <--- Consecutive parts
%e A299775   26   |_ _ _ _ _ _ _|   |   [7, 2]
%e A299775   27   |_ _ _|     |     |   [3, 3, 3]
%e A299775   28   |_ _ _ _ _ _|     |   [6, 3]
%e A299775   29   |_ _ _ _ _|       |   [5, 4]   <--- Consecutive parts
%e A299775   30   |_ _ _ _ _ _ _ _ _|   [9]   <--- Consecutive parts
%e A299775 .
%Y A299775 Row n has length A001227(n).
%Y A299775 Right border gives A000041, n >= 1.
%Y A299775 Cf. A211992 (partitions in colexicographic order).
%Y A299775 Cf. A299765 (partitions into consecutive parts).
%Y A299775 For tables of partitions into consecutive parts see also A286000 and A286001.
%Y A299775 Cf. A000041, A135010, A141285, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A299474, A299475, A299773, A299774.
%K A299775 nonn,more,tabf
%O A299775 1,2
%A A299775 _Omar E. Pol_, Mar 29 2018