A206950 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with no 3-element antichain.

0, 0, 0, 3, 33, 259, 1762, 11093, 66592, 387264, 2202053, 12314587, 67995221, 371697914, 2015659707, 10859379024, 58190011080, 310409500291, 1649579166385, 8738000970251, 46158910515154, 243260704208613, 1279386591175904, 6716811592446952, 35209193397256085

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. Here, the term uniform used in the sense of Retakh, Serconek and Wilson.

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 (13,-59,115,-109,51,-9).

Crossrefs

A206901(n)-A206902(n).

Cf. A206949 (unique maximal element added.)

Cf. A206947, A206948 (requiring exactly two elements in each rank level above 0 with and without maximal element.)

Programs

Mathematica

Join[{0},LinearRecurrence[{13, -59, 115, -109, 51, -9}, {0, 0, 3, 33, 259, 1762}, 40]]

Python

def a(n,adict={0:0,1:0,2:0,3:3,4:33,5:259,6:1762}):
    if n in adict:
        return adict[n]
    adict[n]=13*a(n-1)-59*a(n-2)+115*a(n-3)-109*a(n-4)+51*a(n-5)-9*a(n-6)
    return adict[n]

Formula

a(n) = 13*a(n-1) - 59*a(n-2) + 115*a(n-3) - 109*a(n-4) + 51*a(n-5) - 9*a(n-6), a(1)=0, a(2)=0, a(3)=3, a(4)=33, a(5)=259, a(6)=1762.

G.f.: (3*x^3-6*x^4+7*x^5-3*x^6)/((-1+7*x-10*x^2+3*x^3)*(-1+6*x-7*x^2+3*x^3)).