A206901 Number of nonisomorphic graded posets with 0 of rank n with no 3-element antichain.

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

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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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
• Index entries for linear recurrences with constant coefficients, signature (7,-10,3).

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).

Programs

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}]

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]

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).