cp's OEIS Frontend

A299774 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.

%I A299774 #61 Aug 26 2018 12:31:23
%S A299774 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 A299774 56,1,57,69,73,76,77,1,101,1,102,134,135,1,160,172,176,1,177,221,230,
%U A299774 231,1,297,1,298,353,380,384,385,1,490,1,491,604,615,626,627,1
%N A299774 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 A299774 Note that n is one of the partitions of n into equal parts.
%C A299774 If n is even then row n ending in [p(n) - 1, p(n)], where p(n) = A000041(n).
%C A299774 T(n,k) > p(n - 1), if 1 < k <= A000005(n).
%C A299774 Removing the 1's then all terms of the sequence are in increasing order.
%C A299774 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 A299774 Triangle begins:
%e A299774   1;
%e A299774   1,   2;
%e A299774   1,   3;
%e A299774   1,   4,   5;
%e A299774   1,   7;
%e A299774   1,   8,  10,  11;
%e A299774   1,  15;
%e A299774   1,  16,  21,  22;
%e A299774   1,  27,  30;
%e A299774   1,  31,  41,  42;
%e A299774   1,  56;
%e A299774   1,  57,  69,  73,  76,  77;
%e A299774   1, 101;
%e A299774   1, 102, 134, 135;
%e A299774   1, 160, 172, 176;
%e A299774   ...
%e A299774 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 A299774 -------------------------------------------------------------
%e A299774    p      Diagram        Partitions of 6
%e A299774 -------------------------------------------------------------
%e A299774         _ _ _ _ _ _
%e A299774    1   |_| | | | | |    [1, 1, 1, 1, 1, 1]  <--- equal parts
%e A299774    2   |_ _| | | | |    [2, 1, 1, 1, 1]
%e A299774    3   |_ _ _| | | |    [3, 1, 1, 1]
%e A299774    4   |_ _|   | | |    [2, 2, 1, 1]
%e A299774    5   |_ _ _ _| | |    [4, 1, 1]
%e A299774    6   |_ _ _|   | |    [3, 2, 1]
%e A299774    7   |_ _ _ _ _| |    [5, 1]
%e A299774    8   |_ _|   |   |    [2, 2, 2]  <--- equal parts
%e A299774    9   |_ _ _ _|   |    [4, 2]
%e A299774   10   |_ _ _|     |    [3, 3]  <--- equal parts
%e A299774   11   |_ _ _ _ _ _|    [6]  <--- equal parts
%e A299774 .
%o A299774 (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 A299774 Row n has length A000005(n).
%Y A299774 Right border gives A000041, n >= 1.
%Y A299774 Column 1 gives A000012.
%Y A299774 Records give A317296.
%Y A299774 Cf. A211992 (partitions in colexicographic order).
%Y A299774 Cf. A027750, A135010, A141285, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A299474, A299475, A299773, A299775.
%K A299774 nonn,tabf
%O A299774 1,3
%A A299774 _Omar E. Pol_, Mar 29 2018
%E A299774 Terms a(46) and beyond from _David A. Corneth_, Aug 16 2018