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

0,2

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

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).

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

Adjacent sequences: A206898 A206899 A206900 * A206902 A206903 A206904

Keyword

nonn,easy

Author

David Nacin, Feb 13 2012

Status

approved