A329255 Irregular triangle read by rows: T(n,k) is greatest positive integer <= n that have a partition into k consecutive parts, 1 <= k <= A003056(n), n >= 1.

1, 2, 3, 3, 4, 3, 5, 5, 6, 5, 6, 7, 7, 6, 8, 7, 6, 9, 9, 9, 10, 9, 9, 10, 11, 11, 9, 10, 12, 11, 12, 10, 13, 13, 12, 10, 14, 13, 12, 14, 15, 15, 15, 14, 15, 16, 15, 15, 14, 15, 17, 17, 15, 14, 15, 18, 17, 18, 18, 15, 19, 19, 18, 18, 15, 20, 19, 18, 18, 20, 21, 21, 21, 18, 20, 21, 22, 21, 21, 22, 20, 21

Offset: 1

Author

Omar E. Pol, Nov 09 2019

Keywords

• nonn
• tabf

Comments

T(n,k) is also the positive integer whose partition into k consecutive parts is associated to the k-th vertex, from left to right, of the largest Dyck path of the symmetric representation of sigma(n). For more information see A237593.

Also this triangle can be constructed replacing every zero of triangle A285891 with the previous positive integer from the same column.

Examples

			Triangle begins:
   1;
   2;
   3,  3;
   4,  3;
   5,  5;
   6,  5,  6;
   7,  7,  6;
   8,  7,  6;
   9,  9,  9;
  10,  9,  9, 10;
  11, 11,  9, 10;
  12, 11, 12, 10;
  13, 13, 12, 10;
  14, 13, 12, 14;
  15, 15, 15, 14, 15;
  16, 15, 15, 14, 15;
  17, 17, 15, 14, 15;
  18, 17, 18, 18, 15;
  19, 19, 18, 18, 15;
  20, 19, 18, 18, 20;
  21, 21, 21, 18, 20, 21;
  22, 21, 21, 22, 20, 21;
  23, 23, 21, 22, 20, 21;
  24, 23, 24, 22, 20, 21;
  25, 25, 24, 22, 25, 21;
  26, 25, 24, 26, 25, 21;
  27, 27, 27, 26, 25, 27;
  28, 27, 27, 26, 25, 27, 28;
...
		

Crossrefs

Column k stars with A000217(k) in the row A000217(k).

Row n has length A003056(n).

Cf. A196020, A204217, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A285900, A285914, A285891, A286000, A286001, A286013, A299765, A328361, A328365, A328371.