A222863 Strongly graded (3+1)-free partially ordered sets (posets) on n labeled vertices.

1, 1, 3, 13, 111, 1381, 22383, 461413, 12163791, 420626821, 19880808303, 1337330559973, 130909732781391, 18649561895661061, 3830195104867879023, 1124247654215697637093, 469367653568553278229711, 278046313987470874905216901, 233462156432002170491075384943

Offset: 0

Author

Joel B. Lewis, Mar 07 2013

Keywords

• nonn

Comments

Here "strongly graded" means that every maximal chain has the same length. Alternate terminology includes "graded" (e.g., in Stanley 2012) and "tiered" (as in A006860). A poset is said to be (3+1)-free if it does not contain four elements a, b, c, d such that a < b < c and d is incomparable to the other three.

References

• R. P. Stanley, Enumerative Combinatorics, Volume 1. Cambridge University Press. 2nd edition, 2012. http://math.mit.edu/~rstan/ec/ec1/

Links

• J. B. Lewis and Y. X. Zhang, Enumeration of Graded (3+1)-Avoiding Posets, J. Combin. Theory Ser. A 120 (2013), no. 6, 1305-1327.

Crossrefs

For strongly graded (3+1)-free posets by height, see A222864. For weakly graded (3+1)-free posets, see A222865. For all strongly graded posets, see A006860. For all (3+1)-free posets, see A079145.

Programs

Mathematica

m = maxExponent = 19;
Psi[x_] = Sum[E^(2^n*x)*x^n/n!, {n, 0, m}] + O[x]^m;
H[x_, y_] = 1+(2x^3 - 3x^2 + (x^3 - 4x^2 + 4x)y)/(2x^2 + x + (x^2-2x-1) y);
CoefficientList[H[E^x, Psi[x]] + O[x]^m, x] Range[0, m-1]! (* Jean-François Alcover, Dec 11 2018 *)

Formula

G.f.: H(e^x, Psi(x)) where H(x, y) = 1 + (2x^3 - 3x^2 + (x^3 - 4x^2 + 4x)y)/(2x^2 + x + (x^2 - 2x - 1)y) and Psi(x) is the g.f. for A047863.