A222866 -id:A222866 - cp's OEIS frontend

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

1, 1, 3, 19, 195, 2551, 41343, 826939, 20616795, 658486351, 28264985223, 1725711709459, 155998194920835, 21019550046219271, 4162663551546902223, 1192847436856343300779, 489879387071459457083115, 286844271719979335180726911, 238844671940165660117456403543

Offset: 0

Joel B. Lewis, Mar 07 2013

nonn

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.

J. B. Lewis and Y. X. Zhang, Enumeration of Graded (3+1)-Avoiding Posets, To appear, J. Combinatorial Theory, Series A.

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.

m = maxExponent = 19;
Psi[x_] = Sum[E^(2^n x) x^n/n!, {n, 0, m}] + O[x]^m;
W[x_, y_] = (1-x)y/x + (2x^3 + (x^3 - 2x^2)y)/(2x^2 + x + (x^2-2x-1) y);
CoefficientList[W[E^x, Psi[x]] + O[x]^m, x] Range[0, m-1]! (* Jean-François Alcover, Dec 11 2018 *)

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.

cp's OEIS Frontend

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

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