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.

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, 56, 1, 57, 69, 73, 76, 77, 1, 101, 1, 102, 134, 135, 1, 160, 172, 176, 1, 177, 221, 230, 231, 1, 297, 1, 298, 353, 380, 384, 385, 1, 490, 1, 491, 604, 615, 626, 627, 1

Offset

1,3

Comments

Note that n is one of the partitions of n into equal parts.

If n is even then row n ending in [p(n) - 1, p(n)], where p(n) = A000041(n).

T(n,k) > p(n - 1), if 1 < k <= A000005(n).

Removing the 1's then all terms of the sequence are in increasing order.

If n is even then row n starts with [1, p(n - 1) + 1]. - David A. Corneth and Omar E. Pol, Aug 26 2018

Links

Table of n, a(n) for n=1..67.

Example

Triangle begins:
  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,  56;
  1,  57,  69,  73,  76,  77;
  1, 101;
  1, 102, 134, 135;
  1, 160, 172, 176;
  ...
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].
-------------------------------------------------------------
   p      Diagram        Partitions of 6
-------------------------------------------------------------
        _ _ _ _ _ _
   1   |_| | | | | |    [1, 1, 1, 1, 1, 1]  <--- equal parts
   2   |_ _| | | | |    [2, 1, 1, 1, 1]
   3   |_ _ _| | | |    [3, 1, 1, 1]
   4   |_ _|   | | |    [2, 2, 1, 1]
   5   |_ _ _ _| | |    [4, 1, 1]
   6   |_ _ _|   | |    [3, 2, 1]
   7   |_ _ _ _ _| |    [5, 1]
   8   |_ _|   |   |    [2, 2, 2]  <--- equal parts
   9   |_ _ _ _|   |    [4, 2]
  10   |_ _ _|     |    [3, 3]  <--- equal parts
  11   |_ _ _ _ _ _|    [6]  <--- equal parts
.

Prog

(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

Crossrefs

Row n has length A000005(n).

Right border gives A000041, n >= 1.

Column 1 gives A000012.

Records give A317296.

Cf. A211992 (partitions in colexicographic order).

Cf. A027750, A135010, A141285, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A299474, A299475, A299773, A299775.

Sequence in context: A177687 A219356 A142878 * A362254 A341866 A300242

Adjacent sequences: A299771 A299772 A299773 * A299775 A299776 A299777

Keyword

nonn,tabf

Author

Omar E. Pol, Mar 29 2018

Extensions

Terms a(46) and beyond from David A. Corneth, Aug 16 2018

Status

approved