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A206901
Number of nonisomorphic graded posets with 0 of rank n with no 3-element antichain.
5
1, 2, 8, 39, 199, 1027, 5316, 27539, 142694, 739416, 3831589, 19855045, 102887673, 533158028, 2762794601, 14316644946, 74188042696, 384438233215, 1992137140383, 10323141778619, 53493935746148, 277202543857995, 1436447874880342, 7443591492820888
OFFSET
0,2
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.
LINKS
Vladimir Retakh, Shirlei Serconek, and Robert Wilson, Hilbert series of algebras associated to direct graphs and order homology, arXiv 1010.6295 [math.RA], 2010-2011.
Wikipedia, Graded poset
FORMULA
a(n+3) = 7a(n+2) - 10a(n+1) + 3a(n), a(0)=1, a(1)=2, a(2)=8.
G.f.: (1-5x+4x^2)/(1-7x+10x^2-3x^3).
MATHEMATICA
m = {{3, 3, 1, 0}, {1, 3, 0, 0}, {2, 3, 1, 0}, {6, 9, 2, 0}}; Table[MatrixPower[m, n][[4, 3]], {n, 1, 40}]
PROG
(Python)
def a(n, adict={0:1, 1:2, 2:8}):
if n in adict:
return adict[n]
adict[n]=7*a(n-1)-10*a(n-2)+3*a(n-3)
return adict[n]
CROSSREFS
Cf. A124292 (counts with unique maximal element).
Cf. A025192, A206902 (adding a uniformity condition in the sense of the Retakh et al. paper with and without maximal elements).
Sequence in context: A154133 A077324 A112737 * A162476 A366049 A218321
KEYWORD
nonn,easy
AUTHOR
David Nacin, Feb 13 2012
STATUS
approved