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T(n,k) is the number of labeled connected posets of n labeled elements with k covering relations (n>=1, k>=0). Triangle read by rows.
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%I #10 Mar 16 2021 18:29:11

%S 1,0,2,0,0,12,0,0,0,128,18,0,0,0,0,2000,960,100,0,0,0,0,0,41472,43320,

%T 15000,1710,140,0,0,0,0,0,0,1075648,1985760,1453200,490560,90594,

%U 10080,770,0,0,0,0,0,0,0,33554432,96937680,122360000,82220880,32527488,8205288,1396640,179760,20048,1050

%N T(n,k) is the number of labeled connected posets of n labeled elements with k covering relations (n>=1, k>=0). Triangle read by rows.

%H <a href="/index/Pos#posets">Index entries for sequences related to posets</a>

%e There are 8 connected unlabeled Hasse diagrams on 4 nodes with 3 arcs. 4 of them have automorphism group order 1, 2 of them have automorphism group order 2 and 2 have order 6. So T(4,3) = 4*4!/1 + 2*4!/2 + 2*4!/6 = 128.

%e There are 2 connected unlabeled Hasse diagrams on 4 nodes with 4 arcs, one has automorphism group order 2, the other 4. So T(4,4) = 1*4!/2+1*4!/4 = 18.

%e The triangle starts

%e 1: 1

%e 2: 0 2

%e 3: 0 0 12

%e 4: 0 0 0 128 18

%e 5: 0 0 0 0 2000 960 100

%e 6: 0 0 0 0 0 41472 43320 15000 1710 140

%e 7: 0 0 0 0 0 0 1075648 1985760 1453200 490560 90594 10080 770

%Y Cf. A001927 (row sums), A342589 (not necessarily connected), A342590 (unlabeled).

%K nonn,tabf

%O 1,3

%A _R. J. Mathar_, Mar 16 2021