A206948 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level above 0.

0, 0, 0, 2, 19, 131, 791, 4446, 23913, 124892, 638878, 3218559, 16027375, 79093773, 387540260, 1887974063, 9154751912, 44221373872, 212931964415, 1022594028515, 4900116587043, 23437066655010, 111923110602497

Offset

0,4

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.

References

R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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 (11, -40, 55, -30, 6).

Formula

a(n) = 11*a(n-1) - 40*a(n-2) + 55*a(n-3) - 30*a(n-4) + 6*a(n-5), a(0)=0, a(1)=0, a(2)=0, a(3)=2, a(4)=19, a(5)=131.

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

Mathematica

LinearRecurrence[{11, -40, 55, -30, 6}, {0, 0, 0, 2, 19, 131}, 23] (* David Nacin, Feb 29 2012; a(0) added by Georg Fischer, Apr 03 2019 *)

Prog

(Python)
def a(n, adict={0:0, 1:0, 2:0, 3:2, 4:19, 5:131}):
  if n in adict:
    return adict[n]
  adict[n]=11*a(n-1)-40*a(n-2)+55*a(n-3)-30*a(n-4)+6*a(n-5)
  return adict[n]
for n in range(0, 40):
  print(a(n))

Crossrefs

a(n) = A086405(n) - A012781(n+1).

Cf. A206947 (unique maximal element added).

Cf. A206949, A206950 (allowing one or two elements in each rank level above 0 with and without maximal element).

Sequence in context: A289230 A350568 A249690 * A089364 A178829 A166298

Adjacent sequences: A206945 A206946 A206947 * A206949 A206950 A206951

Keyword

nonn,easy

Author

David Nacin, Feb 13 2012

Status

approved