This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A376894 #24 Feb 08 2025 16:05:05 %S A376894 1,3,14,61,273,1228,5631,26141,123261,589251,2855815,14021038,69707192 %N 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). %C A376894 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 A376894 A342447 is the number of unlabeled posets of j points with k arcs in the Hasse diagram. %C A376894 A342447(j,k)-A342447(j-1,k) = 0 if j > 2k. %C A376894 For k >= 2n-2, A342447(2k-n+1,k)-A342447(2k-n,k) does not depend on k. %C A376894 Therefore we define: a(n) = A342447(2k-n+1,k)-A342447(2k-n,k). %C A376894 A342447(2k-n,k) = A022016(k) - a(1)-...-a(n) for k >= 2n-2, n>0 %C A376894 Proof will soon be submitted to JOIS. %D A376894 R. P. Stanley, Enumerative Combinatorics I, 2nd. ed. %e A376894 See the table of A342447 %e A376894 1 ; %e A376894 1 ; %e A376894 1 1 ; %e A376894 1 1 3 ; %e A376894 1 1 4 8 2 ; %e A376894 1 1 4 11 29 12 5 ; %e A376894 1 1 4 12 43 105 92 45 12 3 ; %e A376894 1 1 4 12 46 156 460 582 487 204 71 14 7 ; %e A376894 1 1 4 12 47 170 670 2097 3822 4514 3271 1579 561 186 44 16 4 ; %e A376894 ... %e A376894 The differences between row j and j-1 of column k (convergence indicated by | |): %e A376894 0 ; %e A376894 0 ; %e A376894 0 |1| ; %e A376894 0 0 |3| ; %e A376894 0 0 |1| 8 2 ; %e A376894 0 0 0 |3| 27 12 5 ; %e A376894 0 0 0 |1| |14| 93 87 45 12 ... ; %e A376894 0 0 0 0 |3| 51 368 537 475 ... ; %e A376894 0 0 0 0 |1| |14| 210 1515 3335 ... ; %e A376894 0 0 0 0 0 |3| |61| 857 6691 ... ; %e A376894 0 0 0 0 0 |1| |14| 258 3683 ... ; %e A376894 0 0 0 0 0 0 |3| |61| 1127 ... ; %e A376894 0 0 0 0 0 0 |1| |14| |273| ... ; %e A376894 a(n) = A342447(2k-n+1,k)-A342447(2k-n,k) for n>=1 %e A376894 e.g. for n = 2 -> k = 2n-2 = 2 %e A376894 a(2) = A342447(3,2) - A342447(2,2) = 3 - 0 = 3 %e A376894 for n = 3 -> k >= 2n-2 = 6 %e A376894 a(3) = A342447(10,6) - A342447(9,6) = 745 - 731 = 14 %Y A376894 Cf. A000112, A022016. %Y A376894 Differences of A342447. %K A376894 nonn,more %O A376894 1,2 %A A376894 Rico Zöllner and _Konrad Handrich_, Oct 22 2024 %E A376894 a(8)-a(13) from _Konrad Handrich_, Jan 07 2025