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).

1, 3, 14, 61, 273, 1228, 5631, 26141, 123261, 589251, 2855815, 14021038, 69707192

Offset: 1

Author

Rico Zöllner and Konrad Handrich, Oct 22 2024

Keywords

• nonn
• more

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.

Examples

			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
		

References

• R. P. Stanley, Enumerative Combinatorics I, 2nd. ed.

Crossrefs

Cf. A000112, A022016.

Differences of A342447.

Extensions

a(8)-a(13) from Konrad Handrich, Jan 07 2025