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

Author

David Nacin, Feb 13 2012

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A025192 (adding a unique maximal element).

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

Programs

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

Magma

R:=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

Mathematica

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

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

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]

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

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).