A206902 Number of nonisomorphic graded posets with 0 and uniform Hasse diagram of rank n with no 3-element antichain.

1, 2, 8, 36, 166, 768, 3554, 16446, 76102, 352152, 1629536, 7540458, 34892452, 161460114, 747134894, 3457265922, 15998031616, 74028732924, 342557973998, 1585140808368, 7335025230994, 33941839649382, 157061283704438, 726779900373936, 3363075935260696

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.

Uniform (in the definition) used in the sense of Retakh, Serconek and Wilson (see paper in Links lines). - David Nacin, Mar 01 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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 (6,-7,3).

Formula

a(n) = 6*a(n-1) - 7*a(n-2) + 3*a(n-3), a(1)=2, a(2)=8, a(3)=36.

G.f.: (1 -4*x +3*x^2 -x^3)/(1 -6*x +7*x^2 -3*x^3).

Mathematica

LinearRecurrence[{6, -7, 3}, {1, 2, 8, 36}, 30] (* Vincenzo Librandi, Feb 27 2012 *)

Prog

(Python)
def a(n, adict={1:2, 2:8, 3:36}):
    if n in adict:
        return adict[n]
    adict[n]=6*a(n-1)-7*a(n-2)+3*a(n-3)
    return adict[n]
(PARI) my(x='x+O('x^30)); Vec((1-4*x+3*x^2-x^3)/(1-6*x+7*x^2-3*x^3)) \\ G. C. Greubel, May 21 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-4*x +3*x^2-x^3)/(1-6*x+7*x^2-3*x^3) )); // G. C. Greubel, May 21 2019
(Sage) ((1-4*x+3*x^2-x^3)/(1-6*x+7*x^2-3*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 21 2019
(GAP) a:=[2, 8, 36];; for n in [4..30] do a[n]:=6*a[n-1]-7*a[n-2]+3*a[n-3]; od; Concatenation([1], a); # G. C. Greubel, May 21 2019

Crossrefs

Cf. A025192 (adding a unique maximal element).

Cf. A124292, A206901 (dropping uniformity with and without maximal element).

Sequence in context: A152124 A147722 A089387 * A275752 A084868 A350645

Adjacent sequences: A206899 A206900 A206901 * A206903 A206904 A206905

Keyword

nonn,easy

Author

David Nacin, Feb 13 2012

Status

approved