login
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.
5
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
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) = 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: A086243 A375874 A258138 * A203241 A072888 A171012
KEYWORD
nonn,easy
AUTHOR
David Nacin, Feb 13 2012
STATUS
approved