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

0,7

References

Maximal e for a given n (i.e., the length of the n-th row minus 1) is no less than A002620(n) and possibly equals that value. - Andrey Zabolotskiy, Mar 12 2021

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

Formula

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

T(n,e) = A022016(e) for n >= 2e. - Andrey Zabolotskiy, Mar 12 2021

Example

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

Crossrefs

Cf. A000112 (row sums), A263864, A022016 (convergents down rows), A002620, A342472 (lower bound row length).

Sequence in context: A263950 A160870 A345279 * A025255 A296006 A245397

Adjacent sequences: A342444 A342445 A342446 * A342448 A342449 A342450

Keyword

nonn,tabf,hard

Author

R. J. Mathar, Mar 12 2021

Extensions

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

Status

approved