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