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

2,2

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.

Links

Table of n, a(n) for n=2..56.

Example

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).

Sequence in context: A160185 A283424 A188392 * A143409 A197387 A171177

Adjacent sequences: A356555 A356556 A356557 * A356559 A356560 A356561

Keyword

nonn,tabl,more

Author

Salah Uddin Mohammad, Aug 12 2022

Status

approved