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%I #61 Aug 26 2018 12:31:23
%S 1,1,2,1,3,1,4,5,1,7,1,8,10,11,1,15,1,16,21,22,1,27,30,1,31,41,42,1,
%T 56,1,57,69,73,76,77,1,101,1,102,134,135,1,160,172,176,1,177,221,230,
%U 231,1,297,1,298,353,380,384,385,1,490,1,491,604,615,626,627,1
%N Irregular triangle read by rows in which row n lists the indices of the partitions into equal parts in the list of colexicographically ordered partitions of n.
%C Note that n is one of the partitions of n into equal parts.
%C If n is even then row n ending in [p(n) - 1, p(n)], where p(n) = A000041(n).
%C T(n,k) > p(n - 1), if 1 < k <= A000005(n).
%C Removing the 1's then all terms of the sequence are in increasing order.
%C If n is even then row n starts with [1, p(n - 1) + 1]. - _David A. Corneth_ and _Omar E. Pol_, Aug 26 2018
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 3;
%e 1, 4, 5;
%e 1, 7;
%e 1, 8, 10, 11;
%e 1, 15;
%e 1, 16, 21, 22;
%e 1, 27, 30;
%e 1, 31, 41, 42;
%e 1, 56;
%e 1, 57, 69, 73, 76, 77;
%e 1, 101;
%e 1, 102, 134, 135;
%e 1, 160, 172, 176;
%e ...
%e For n = 6 the partitions of 6 into equal parts are [1, 1, 1, 1, 1, 1], [2, 2, 2], [3, 3] and [6]. Then we have that in the list of colexicographically ordered partitions of 6 these partitions are in the rows 1, 8, 10 and 11 respectively as shown below, so the 6th row of the triangle is [1, 8, 10, 11].
%e -------------------------------------------------------------
%e p Diagram Partitions of 6
%e -------------------------------------------------------------
%e _ _ _ _ _ _
%e 1 |_| | | | | | [1, 1, 1, 1, 1, 1] <--- equal parts
%e 2 |_ _| | | | | [2, 1, 1, 1, 1]
%e 3 |_ _ _| | | | [3, 1, 1, 1]
%e 4 |_ _| | | | [2, 2, 1, 1]
%e 5 |_ _ _ _| | | [4, 1, 1]
%e 6 |_ _ _| | | [3, 2, 1]
%e 7 |_ _ _ _ _| | [5, 1]
%e 8 |_ _| | | [2, 2, 2] <--- equal parts
%e 9 |_ _ _ _| | [4, 2]
%e 10 |_ _ _| | [3, 3] <--- equal parts
%e 11 |_ _ _ _ _ _| [6] <--- equal parts
%e .
%o (PARI) row(n) = {if(n == 1, return([1])); my(nd = numdiv(n), res = vector(nd)); res[1] = 1; res[nd] = numbpart(n); if(nd > 2, t = nd - 1; p = vecsort(partitions(n)); forstep(i = #p - 1, 2, -1, if(p[i][1] == p[i][#p[i]], res[t] = i; t--; if(t==1, return(res)))), return(res))} \\ _David A. Corneth_, Aug 17 2018
%Y Row n has length A000005(n).
%Y Right border gives A000041, n >= 1.
%Y Column 1 gives A000012.
%Y Records give A317296.
%Y Cf. A211992 (partitions in colexicographic order).
%Y Cf. A027750, A135010, A141285, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A299474, A299475, A299773, A299775.
%K nonn,tabf
%O 1,3
%A _Omar E. Pol_, Mar 29 2018
%E Terms a(46) and beyond from _David A. Corneth_, Aug 16 2018
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