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A342447 T(n,e) is the number of unlabeled posets of n>=0 points with e>=0 arcs in the Hasse diagram, irregular triangle read by rows.

%I A342447 #36 Oct 19 2024 21:26:47
%S A342447 1,1,1,1,1,1,3,1,1,4,8,2,1,1,4,11,29,12,5,1,1,4,12,43,105,92,45,12,3,
%T A342447 1,1,4,12,46,156,460,582,487,204,71,14,7,1,1,4,12,47,170,670,2097,
%U A342447 3822,4514,3271,1579,561,186,44,16,4,1,1,4,12,47,173,731,2954,10513,24584,40182
%N A342447 T(n,e) is the number of unlabeled posets of n>=0 points with e>=0 arcs in the Hasse diagram, irregular triangle read by rows.
%C A342447 Maximal e for a given n (i.e., the length of the n-th row minus 1) is A002620(n), see Mathematics StackExchange. - _Andrey Zabolotskiy_, Mar 12 2021
%H A342447 R. J. Mathar, <a href="/A342447/b342447.txt">Table of T(n,e) for n<=10</a>
%H A342447 Brendan McKay, <a href="http://users.cecs.anu.edu.au/~bdm/data/digraphs.html">Digraphs</a>, posets tables at the bottom of the page.
%H A342447 R. J. Mathar, <a href="/A342447/a342447.pdf">Poset Illustrations</a>
%H A342447 Mathematics StackExchange, <a href="https://math.stackexchange.com/questions/4190312">Maximum number of comparisons required to define a partial ordering</a>, 2021.
%F A342447 T(n,0) = T(n,1) = 1.
%F A342447 T(n,e) = A022016(e) for n >= 2e.
%e A342447 The table starts
%e A342447 1 ;
%e A342447 1 ;
%e A342447 1 1 ;
%e A342447 1 1 3 ;
%e A342447 1 1 4  8  2 ;
%e A342447 1 1 4 11 29  12   5 ;
%e A342447 1 1 4 12 43 105  92   45   12    3 ;
%e A342447 1 1 4 12 46 156 460  582  487  204   71   14   7 ;
%e A342447 1 1 4 12 47 170 670 2097 3822 4514 3271 1579 561 186 44 16 4 ;
%e A342447 ...
%e A342447 T(4,0) = 1: the 4-point poset with no relations, 4 isolated points in the Hasse diagram.
%e A342447 T(4,1) = 1: the 4-point poset with one relation, the Hasse diagram has one vertical line and 2 isolated points.
%e A342447 T(4,2) = 4: the 4 posets contributing to A022016(4) = 4, extended by additional isolated point when the number of points is less than 4.
%e A342447 T(4,3) = 8: the 8 posets contributing to A022017(3).
%e A342447 T(4,4) = 2: the "dagaz rune" poset {1<3, 2<3, 1<4, 2<4}
%e A342447   o o
%e A342447   |X|
%e A342447   o o
%e A342447 and the "diamond" poset {1<2, 1<3, 2<4, 3<4}
%e A342447     o
%e A342447    / \
%e A342447   o   o
%e A342447    \ /
%e A342447     o
%Y A342447 Cf. A000112 (row sums), A263864, A022016 (convergents down rows), A002620, A342472 (lower bound row length), A342590 (connected), A342589 (labeled), A376633 (self-dual).
%K A342447 nonn,tabf,hard
%O A342447 0,7
%A A342447 _R. J. Mathar_, Mar 12 2021
%E A342447 T(0,0) = 1 prepended and "conjecture" removed from A022016 formula. _Andrey Zabolotskiy_, Mar 12 2021