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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.

%I A356558 #13 Oct 05 2022 04:46:37
%S A356558 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,
%T A356558 27,7,1,2638,1523,744,315,116,35,8,1,10272,5934,2908,1261,483,161,44,
%U A356558 9,1,40782,23505,11580,5085,2010,707,216,54,10,1
%N 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.
%C A356558 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.
%e A356558 Triangle begins:
%e A356558       1;
%e A356558       2,     1;
%e A356558       5,     3,     1;
%e A356558      16,     9,     4,    1;
%e A356558      52,    31,    14,    5,    1;
%e A356558     188,   108,    52,   20,    6,   1;
%e A356558     690,   402,   193,   80,   27,   7,   1;
%e A356558    2638,  1523,   744,  315,  116,  35,   8,  1;
%e A356558   10272,  5934,  2908, 1261,  483, 161,  44,  9,  1;
%e A356558   40782, 23505, 11580, 5085, 2010, 707, 216, 54, 10, 1;
%e A356558 The connected posets counted in the first three rows of the triangle are shown by using the Hasse diagram as follows:
%e A356558 -------
%e A356558   o
%e A356558   |
%e A356558   o
%e A356558 --------------------------
%e A356558                   |   o
%e A356558     o     o   o   |   |
%e A356558    / \     \ /    |   o
%e A356558   o   o     o     |   |
%e A356558                   |   o
%e A356558 ----------------------------------------------------------
%e A356558     o    o o o   o o    |                           |
%e A356558    /|\    \|/    |X|    |                           |   o
%e A356558   o o o    o     o o    |     o     o   o     o     |   |
%e A356558                         |     |      \ /     / \    |   o
%e A356558     o           o       |     o       o     o   o   |   |
%e A356558     |          / \      |    / \      |      \ /    |   o
%e A356558     o   o     o   \     |   o   o     o       o     |   |
%e A356558      \ /      |    \    |                           |   o
%e A356558       o       o     o   |                           |
%Y A356558 Row sums give A007453.
%Y A356558 Cf. A263864 (all posets), A349488 (disconnected).
%K A356558 nonn,tabl,more
%O A356558 2,2
%A A356558 _Salah Uddin Mohammad_, Aug 12 2022