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.

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, 1, 1, 4, 12, 46, 156, 460, 582, 487, 204, 71, 14, 7, 1, 1, 4, 12, 47, 170, 670, 2097, 3822, 4514, 3271, 1579, 561, 186, 44, 16, 4, 1, 1, 4, 12, 47, 173, 731, 2954, 10513, 24584, 40182

Offset: 0

Author

R. J. Mathar, Mar 12 2021

Keywords

• nonn
• tabf
• hard

Comments

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

Examples

			The table starts
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 ;
1 1 4 12 46 156 460  582  487  204   71   14   7 ;
1 1 4 12 47 170 670 2097 3822 4514 3271 1579 561 186 44 16 4 ;
...
T(4,0) = 1: the 4-point poset with no relations, 4 isolated points in the Hasse diagram.
T(4,1) = 1: the 4-point poset with one relation, the Hasse diagram has one vertical line and 2 isolated points.
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.
T(4,3) = 8: the 8 posets contributing to A022017(3).
T(4,4) = 2: the "dagaz rune" poset {1<3, 2<3, 1<4, 2<4}
  o o
  |X|
  o o
and the "diamond" poset {1<2, 1<3, 2<4, 3<4}
    o
   / \
  o   o
   \ /
    o
		

Links

• R. J. Mathar, Table of T(n,e) for n<=10
• Brendan McKay, Digraphs, posets tables at the bottom of the page.
• R. J. Mathar, Poset Illustrations
• Mathematics StackExchange, Maximum number of comparisons required to define a partial ordering, 2021.

Crossrefs

Cf. A000112 (row sums), A263864, A022016 (convergents down rows), A002620, A342472 (lower bound row length), A342590 (connected), A342589 (labeled), A376633 (self-dual).

Formula

T(n,0) = T(n,1) = 1.

T(n,e) = A022016(e) for n >= 2e.

Extensions

T(0,0) = 1 prepended and "conjecture" removed from A022016 formula. Andrey Zabolotskiy, Mar 12 2021