OFFSET
1,2
COMMENTS
Number of unlabeled posets A342447(j,k) with j points, without isolated points, with k arcs in the Hasse diagramm missing n points to achieve saturation of the poset i.e. j=2k-n+1.
A342447 is the number of unlabeled posets of j points with k arcs in the Hasse diagram.
Proof will soon be submitted to JOIS.
REFERENCES
R. P. Stanley, Enumerative Combinatorics I, 2nd. ed.
EXAMPLE
See the table of 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 ;
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 ;
...
The differences between row j and j-1 of column k (convergence indicated by | |):
0 ;
0 ;
0 |1| ;
0 0 |3| ;
0 0 |1| 8 2 ;
0 0 0 |3| 27 12 5 ;
0 0 0 |1| |14| 93 87 45 12 ... ;
0 0 0 0 |3| 51 368 537 475 ... ;
0 0 0 0 |1| |14| 210 1515 3335 ... ;
0 0 0 0 0 |3| |61| 857 6691 ... ;
0 0 0 0 0 |1| |14| 258 3683 ... ;
0 0 0 0 0 0 |3| |61| 1127 ... ;
0 0 0 0 0 0 |1| |14| |273| ... ;
e.g. for n = 2 -> k = 2n-2 = 2
for n = 3 -> k >= 2n-2 = 6
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Rico Zöllner and Konrad Handrich, Oct 22 2024
EXTENSIONS
a(8)-a(13) from Konrad Handrich, Jan 07 2025
STATUS
approved