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A206950
Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with no 3-element antichain.
6
0, 0, 0, 3, 33, 259, 1762, 11093, 66592, 387264, 2202053, 12314587, 67995221, 371697914, 2015659707, 10859379024, 58190011080, 310409500291, 1649579166385, 8738000970251, 46158910515154, 243260704208613, 1279386591175904, 6716811592446952, 35209193397256085
OFFSET
0,4
COMMENTS
We do not assume all maximal elements have maximal rank and thus use graded poset to mean: For every element x, all maximal chains among those with x as greatest element have the same finite length. Here, the term uniform used in the sense of Retakh, Serconek and Wilson.
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) = 13*a(n-1) - 59*a(n-2) + 115*a(n-3) - 109*a(n-4) + 51*a(n-5) - 9*a(n-6), a(1)=0, a(2)=0, a(3)=3, a(4)=33, a(5)=259, a(6)=1762.
G.f.: (3*x^3-6*x^4+7*x^5-3*x^6)/((-1+7*x-10*x^2+3*x^3)*(-1+6*x-7*x^2+3*x^3)).
MATHEMATICA
Join[{0}, LinearRecurrence[{13, -59, 115, -109, 51, -9}, {0, 0, 3, 33, 259, 1762}, 40]]
PROG
(Python)
def a(n, adict={0:0, 1:0, 2:0, 3:3, 4:33, 5:259, 6:1762}):
if n in adict:
return adict[n]
adict[n]=13*a(n-1)-59*a(n-2)+115*a(n-3)-109*a(n-4)+51*a(n-5)-9*a(n-6)
return adict[n]
CROSSREFS
Cf. A206949 (unique maximal element added.)
Cf. A206947, A206948 (requiring exactly two elements in each rank level above 0 with and without maximal element.)
Sequence in context: A195578 A370083 A331193 * A189644 A003129 A190542
KEYWORD
nonn,easy
AUTHOR
David Nacin, Feb 13 2012
STATUS
approved