A356558 Triangle read by rows: T(n,k), where n, k >= 2, is the number of n-element unlabeled connected series-parallel posets with k ordinal terms that are either the singleton or disconnected posets.

1, 2, 1, 5, 3, 1, 16, 9, 4, 1, 52, 31, 14, 5, 1, 188, 108, 52, 20, 6, 1, 690, 402, 193, 80, 27, 7, 1, 2638, 1523, 744, 315, 116, 35, 8, 1, 10272, 5934, 2908, 1261, 483, 161, 44, 9, 1, 40782, 23505, 11580, 5085, 2010, 707, 216, 54, 10, 1

Offset: 2

Author

Salah Uddin Mohammad, Aug 12 2022

Keywords

• nonn
• tabl
• more

Comments

If a poset P is obtained by taking the ordinal sum of the posets A and B, then the posets A and B are called the ordinal terms of P.

Examples

			Triangle begins:
      1;
      2,     1;
      5,     3,     1;
     16,     9,     4,    1;
     52,    31,    14,    5,    1;
    188,   108,    52,   20,    6,   1;
    690,   402,   193,   80,   27,   7,   1;
   2638,  1523,   744,  315,  116,  35,   8,  1;
  10272,  5934,  2908, 1261,  483, 161,  44,  9,  1;
  40782, 23505, 11580, 5085, 2010, 707, 216, 54, 10, 1;
The connected posets counted in the first three rows of the triangle are shown by using the Hasse diagram as follows:
-------
  o
  |
  o
--------------------------
                  |   o
    o     o   o   |   |
   / \     \ /    |   o
  o   o     o     |   |
                  |   o
----------------------------------------------------------
    o    o o o   o o    |                           |
   /|\    \|/    |X|    |                           |   o
  o o o    o     o o    |     o     o   o     o     |   |
                        |     |      \ /     / \    |   o
    o           o       |     o       o     o   o   |   |
    |          / \      |    / \      |      \ /    |   o
    o   o     o   \     |   o   o     o       o     |   |
     \ /      |    \    |                           |   o
      o       o     o   |                           |
		

Crossrefs

Row sums give A007453.

Cf. A263864 (all posets), A349488 (disconnected).