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

0,4

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

Table of n, a(n) for n=0..26.

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).

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).

Mathematica

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

Prog

(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]

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.)

Sequence in context: A084770 A086243 A258138 * A203241 A072888 A171012

Adjacent sequences: A206944 A206945 A206946 * A206948 A206949 A206950

Keyword

nonn,easy

Author

David Nacin, Feb 13 2012

Status

approved