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A206949
Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with no 3-element antichain.
4
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
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
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
KEYWORD
nonn,easy
AUTHOR
David Nacin, Feb 13 2012
STATUS
approved