A206949 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with no 3-element antichain.

0, 0, 0, 3, 24, 135, 657, 2961, 12744, 53244, 218025, 880308, 3518721, 13961727, 55097091, 216546048, 848476296, 3316800555, 12942852624, 50437433079, 196347606849, 763752142233, 2969021213928, 11536374392820, 44809232564673, 173997851613660, 675501426136017

Offset: 0

Author

David Nacin, Feb 13 2012

Keywords

• nonn
• easy

Comments

Here, the term uniform is used in the sense of Retakh, Serconek and Wilson. Graded is used in terms of Stanley's definition that all maximal chains have the same length n.

References

• Richard P. 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 (9,-27,30,-9).

Crossrefs

Cf. A206950 (maximal element removed).

Cf. A206947, A206948 (requiring exactly two elements in each rank level above 0 with and without maximal element).

Programs

Mathematica

Join[{0}, LinearRecurrence[{9, -27, 30, -9}, {0, 0, 3, 24}, 40]]

Python

def a(n,adict={0:0,1:0,2:0,3:3,4:24}):
    if n in adict:
        return adict[n]
    adict[n]=9*a(n-1)-27*a(n-2)+30*a(n-3)-9*a(n-4)
    return adict[n]

Formula

a(n) = 9*a(n-1) - 27*a(n-2) + 30*a(n-3) - 9*a(n-4), a(1)=0, a(2)=0, a(3)=3, a(4)=24.

G.f.: (3*(1-x)*x^3)/((1-3*x)*(1-6*x+9*x^2-3*x^3)).

a(n) = A124292(n+1) - A025192(n).