A206947 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank above 0.

0, 0, 0, 2, 14, 70, 306, 1248, 4888, 18666, 70110, 260414, 959882, 3519232, 12854064, 46824210, 170243566, 618125238, 2242100898, 8126927456, 29442587720, 106626616954, 386046638142, 1397431266222, 5057790129274, 18304064121600, 66237312391776

Offset: 0

Author

David Nacin, Feb 13 2012

Keywords

• nonn
• easy

Comments

Here, the term uniform 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

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

Links

• 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 (8,-21,20,-5).

Crossrefs

Cf. A206948 (removing unique maximal element.)

Cf. A206949, A206950 (allowing one or two elements in each rank level above 0 with and without maximal element.)

Programs

Mathematica

Join[{0}, LinearRecurrence[{8, -21, 20, -5}, {0, 0, 2, 14}, 40]]

Python

def a(n,adict={0:0,1:0,2:0,3:2,4:14}):
    if n in adict:
        return adict[n]
    adict[n]=8*a(n-1)-21*a(n-2)+20*a(n-3)-5*a(n-4)
    return adict[n]

Formula

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), a(1)=0, a(2)=0, a(3)=2, a(4)=14.

G.f.: (2*(1-x)*x^3)/((1-3*x+x^2)*(1-5*x+5*x^2)).

a(n) = A081567(n-1) - A001906(n).