A208737 - OEIS

A208737

Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with no 3-element antichain.

0, 0, 0, 1, 7, 37, 175, 778, 3325, 13837, 56524, 227866, 909832, 3607294, 14227447, 55894252, 218937532, 855650749, 3338323915, 13007422705, 50631143323, 196928737582, 765495534433, 2974251390529, 11552064922624, 44856304154086

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Uniform used in the sense of Retakh, Serconek and Wilson. We use Stanley's definition of graded poset: all maximal chains have the same length n (which also implies all maximal elements have maximal rank.)

REFERENCES

R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.

Wikipedia, Graded poset

Index entries for linear recurrences with constant coefficients, signature (10,-36,57,-39,9).

FORMULA

a(n) = 10*a(n-1) - 36*a(n-2) + 57*a(n-3) - 39*a(n-4) + 9*a(n-5), a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 7, a(5) = 37.

G.f: (x^3 - 3*x^4 + 3*x^5)/(1 - 10*x + 36*x^2 - 57*x^3 + 39*x^4 - 9*x^5); (x^3*(1 - 3*x + 3*x^2)) / ((1 - x) (1 - 3*x) (1 - 6*x + 9*x^2 - 3*x^3)).

a(n) = A124292(n) - A124302(n).

MATHEMATICA