A376894 - OEIS

%I #24 Feb 08 2025 16:05:05

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

%N 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).

%C 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.

%C A342447 is the number of unlabeled posets of j points with k arcs in the Hasse diagram.

%C A342447(j,k)-A342447(j-1,k) = 0 if j > 2k.

%C For k >= 2n-2, A342447(2k-n+1,k)-A342447(2k-n,k) does not depend on k.

%C Therefore we define: a(n) = A342447(2k-n+1,k)-A342447(2k-n,k).

%C A342447(2k-n,k) = A022016(k) - a(1)-...-a(n) for k >= 2n-2, n>0

%C Proof will soon be submitted to JOIS.

%D R. P. Stanley, Enumerative Combinatorics I, 2nd. ed.

%e See the table of A342447

%e  1 ;

%e  1 ;

%e  1 1 ;

%e  1 1 3 ;

%e  1 1 4  8  2 ;

%e  1 1 4 11 29  12   5 ;

%e  1 1 4 12 43 105  92   45   12    3 ;

%e  1 1 4 12 46 156 460  582  487  204   71   14   7 ;

%e  1 1 4 12 47 170 670 2097 3822 4514 3271 1579 561 186 44 16 4 ;

%e  ...

%e The differences between row j and j-1 of column k (convergence indicated by | |):

%e  0 ;

%e  0 ;

%e  0 |1| ;

%e  0  0 |3| ;

%e  0  0 |1| 8    2 ;

%e  0  0  0 |3|  27    12     5 ;

%e  0  0  0 |1| |14|   93    87      45    12   ... ;

%e  0  0  0  0   |3|   51   368     537   475   ... ;

%e  0  0  0  0   |1|  |14|  210    1515  3335   ... ;

%e  0  0  0  0    0    |3|  |61|    857  6691   ... ;

%e  0  0  0  0    0    |1|  |14|    258  3683   ... ;

%e  0  0  0  0    0     0    |3|    |61| 1127   ... ;

%e  0  0  0  0    0     0    |1|    |14| |273|  ... ;

%e a(n) = A342447(2k-n+1,k)-A342447(2k-n,k) for n>=1

%e e.g. for n = 2 -> k = 2n-2 = 2

%e a(2) = A342447(3,2) - A342447(2,2) = 3 - 0 = 3

%e for n = 3 -> k >= 2n-2 = 6

%e a(3) = A342447(10,6) - A342447(9,6) = 745 - 731 = 14

%Y Cf. A000112, A022016.

%Y Differences of A342447.

%K nonn,more

%O 1,2

%A Rico Zöllner and _Konrad Handrich_, Oct 22 2024

%E a(8)-a(13) from _Konrad Handrich_, Jan 07 2025