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

0,4

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).

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).

Mathematica

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

Prog

(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]

Crossrefs

Cf. A206950 (maximal element removed).

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

Sequence in context: A001089 A359884 A069515 * A215636 A056350 A056344

Adjacent sequences: A206946 A206947 A206948 * A206950 A206951 A206952

Keyword

nonn,easy

Author

David Nacin, Feb 13 2012

Status

approved