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A206948
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Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level above 0.
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4
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0, 0, 0, 2, 19, 131, 791, 4446, 23913, 124892, 638878, 3218559, 16027375, 79093773, 387540260, 1887974063, 9154751912, 44221373872, 212931964415, 1022594028515, 4900116587043, 23437066655010, 111923110602497
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OFFSET
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0,4
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COMMENTS
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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.
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REFERENCES
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R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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 (11, -40, 55, -30, 6).
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FORMULA
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a(n) = 11*a(n-1) - 40*a(n-2) + 55*a(n-3) - 30*a(n-4) + 6*a(n-5), a(0)=0, a(1)=0, a(2)=0, a(3)=2, a(4)=19, a(5)=131.
G.f.: (x^3*(2 - 3*x + 2*x^2))/((1 - 6*x + 6*x^2)*(1 - 5*x + 4*x^2 - x^3)).
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MATHEMATICA
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LinearRecurrence[{11, -40, 55, -30, 6}, {0, 0, 0, 2, 19, 131}, 23] (* David Nacin, Feb 29 2012; a(0) added by Georg Fischer, Apr 03 2019 *)
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PROG
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(Python)
def a(n, adict={0:0, 1:0, 2:0, 3:2, 4:19, 5:131}):
if n in adict:
return adict[n]
adict[n]=11*a(n-1)-40*a(n-2)+55*a(n-3)-30*a(n-4)+6*a(n-5)
return adict[n]
for n in range(0, 40):
print(a(n))
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CROSSREFS
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a(n) = A086405(n) - A012781(n+1).
Cf. A206947 (unique maximal element added).
Cf. A206949, A206950 (allowing one or two elements in each rank level above 0 with and without maximal element).
Sequence in context: A289230 A350568 A249690 * A089364 A178829 A166298
Adjacent sequences: A206945 A206946 A206947 * A206949 A206950 A206951
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KEYWORD
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nonn,easy
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AUTHOR
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David Nacin, Feb 13 2012
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STATUS
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approved
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