A208736 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level between 0 and 1.

0, 0, 0, 1, 5, 22, 91, 361, 1392, 5265, 19653, 72694, 267179, 977593, 3565600, 12975457, 47142021, 171075606, 620303547, 2247803785, 8141857808, 29481675889, 106728951109, 386314552438, 1398132674955, 5059626441177, 18308871648576, 66249898660801

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 (8,-21,20,-5).

Formula

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), a(2) = 0, a(3) = 1, a(4) = 5, a(5) = 22.

G.f.: (x^3 - 3*x^4 + 3*x^5)/(1 - 8*x + 21*x^2 - 20*x^3 + 5*x^4); (x^3 * (1 - 3*x + 3*x^2))/((1 - 3*x + x^2)*(1 - 5*x + 5*x^2)) .

a(n) = A081567(n-2) - A001519(n-1).

Mathematica

Join[{0, 0}, LinearRecurrence[{8, -21, 20, -5}, {0, 1, 5, 22}, 40]]

Prog

(Python)
def a(n, d={0:0, 1:0, 2:0, 3:1, 4:5, 5:22}):
    if n in d:
        return d[n]
    d[n]=8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4)
    return d[n]

Crossrefs

Cf. A208737, A206901, A206902, A206947-A206950, A001906, A025192, A081567, A124302, A124292, A088305, A086405, A012781.

Sequence in context: A128566 A097138 A105467 * A050185 A216597 A085812

Adjacent sequences: A208733 A208734 A208735 * A208737 A208738 A208739

Keyword

nonn,easy

Author

David Nacin, Mar 01 2012

Status

approved