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%I #14 Mar 11 2024 23:11:15
%S 1,1,3,19,195,2551,41343,826939,20616795,658486351,28264985223,
%T 1725711709459,155998194920835,21019550046219271,4162663551546902223,
%U 1192847436856343300779,489879387071459457083115,286844271719979335180726911,238844671940165660117456403543
%N Weakly graded (3+1)-free partially ordered sets (posets) on n labeled vertices.
%C 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 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 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 m = maxExponent = 19;
%t Psi[x_] = Sum[E^(2^n x) x^n/n!, {n, 0, m}] + O[x]^m;
%t W[x_, y_] = (1-x)y/x + (2x^3 + (x^3 - 2x^2)y)/(2x^2 + x + (x^2-2x-1) y);
%t CoefficientList[W[E^x, Psi[x]] + O[x]^m, x] Range[0, m-1]! (* _Jean-François Alcover_, Dec 11 2018 *)
%Y 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 nonn
%O 0,3
%A _Joel B. Lewis_, Mar 07 2013
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