cp's OEIS Frontend

A222865 Weakly graded (3+1)-free partially ordered sets (posets) on n labeled vertices.

%I A222865 #14 Mar 11 2024 23:11:15
%S A222865 1,1,3,19,195,2551,41343,826939,20616795,658486351,28264985223,
%T A222865 1725711709459,155998194920835,21019550046219271,4162663551546902223,
%U A222865 1192847436856343300779,489879387071459457083115,286844271719979335180726911,238844671940165660117456403543
%N A222865 Weakly graded (3+1)-free partially ordered sets (posets) on n labeled vertices.
%C A222865 Here "weakly graded" means that there is a rank function rk from the vertices to the integers such that whenever x covers y we have rk(x) = rk(y) + 1.  Alternate terminology includes "graded" and "ranked."  A poset is said to be (3+1)-free if it does not contain four vertices  a, b, c, d such that a < b < c and d is incomparable to the other three.
%H A222865 J. B. Lewis and Y. X. Zhang, <a href="http://arxiv.org/abs/1106.5480">Enumeration of Graded (3+1)-Avoiding Posets</a>, To appear, J. Combinatorial Theory, Series A.
%F A222865 G.f. is W(e^x, Psi(x)) where W(x, y) = (1 - x)y/x + (2x^3 + (x^3 - 2x^2)y)/(2x^2 + x + (x^2 - 2x - 1)y) and Psi(x) is the GF for A047863.
%t A222865 m = maxExponent = 19;
%t A222865 Psi[x_] = Sum[E^(2^n x) x^n/n!, {n, 0, m}] + O[x]^m;
%t A222865 W[x_, y_] = (1-x)y/x + (2x^3 + (x^3 - 2x^2)y)/(2x^2 + x + (x^2-2x-1) y);
%t A222865 CoefficientList[W[E^x, Psi[x]] + O[x]^m, x] Range[0, m-1]! (* _Jean-François Alcover_, Dec 11 2018 *)
%Y A222865 For weakly graded (3+1)-free posets by height, see A222866.  For strongly graded (3+1)-free posets, see A222863.  For all weakly graded posets, see A001833.  For all (3+1)-free posets, see A079145.
%K A222865 nonn
%O A222865 0,3
%A A222865 _Joel B. Lewis_, Mar 07 2013