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A376894
Stationary differences in A342447: a(n) = A342447(2k-n+1,k)-A342447(2k-n,k) which does not depend on k if k>= 2n-2 (for n>0).
0
1, 3, 14, 61, 273, 1228, 5631, 26141, 123261, 589251, 2855815, 14021038, 69707192
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.
A342447(j,k)-A342447(j-1,k) = 0 if j > 2k.
For k >= 2n-2, A342447(2k-n+1,k)-A342447(2k-n,k) does not depend on k.
Therefore we define: a(n) = A342447(2k-n+1,k)-A342447(2k-n,k).
A342447(2k-n,k) = A022016(k) - a(1)-...-a(n) for k >= 2n-2, n>0
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| ... ;
a(n) = A342447(2k-n+1,k)-A342447(2k-n,k) for n>=1
e.g. for n = 2 -> k = 2n-2 = 2
a(2) = A342447(3,2) - A342447(2,2) = 3 - 0 = 3
for n = 3 -> k >= 2n-2 = 6
a(3) = A342447(10,6) - A342447(9,6) = 745 - 731 = 14
CROSSREFS
Differences of A342447.
Sequence in context: A307268 A237608 A100295 * A291025 A320499 A091701
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