A206948 -id:A206948 - cp's OEIS frontend

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

David Nacin, Feb 13 2012

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.

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).

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.)

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

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]

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)).

A206947 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank above 0.

Original entry on oeis.org

0, 0, 0, 2, 14, 70, 306, 1248, 4888, 18666, 70110, 260414, 959882, 3519232, 12854064, 46824210, 170243566, 618125238, 2242100898, 8126927456, 29442587720, 106626616954, 386046638142, 1397431266222, 5057790129274, 18304064121600, 66237312391776

Offset: 0

David Nacin, Feb 13 2012

Here, the term uniform used in the sense of Retakh, Serconek and Wilson. Graded is used in terms of Stanley's definition that all maximal chains have the same length n.

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

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 (8,-21,20,-5).

Cf. A206948 (removing unique maximal element.)

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

Join[{0}, LinearRecurrence[{8, -21, 20, -5}, {0, 0, 2, 14}, 40]]

def a(n,adict={0:0,1:0,2:0,3:2,4:14}):
    if n in adict:
        return adict[n]
    adict[n]=8*a(n-1)-21*a(n-2)+20*a(n-3)-5*a(n-4)
    return adict[n]

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), a(1)=0, a(2)=0, a(3)=2, a(4)=14.
G.f.: (2*(1-x)*x^3)/((1-3*x+x^2)*(1-5*x+5*x^2)).
a(n) = A081567(n-1) - A001906(n).

A206949 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with no 3-element antichain.

Original entry on oeis.org

0, 0, 0, 3, 24, 135, 657, 2961, 12744, 53244, 218025, 880308, 3518721, 13961727, 55097091, 216546048, 848476296, 3316800555, 12942852624, 50437433079, 196347606849, 763752142233, 2969021213928, 11536374392820, 44809232564673, 173997851613660, 675501426136017

Offset: 0

David Nacin, Feb 13 2012

Here, the term uniform is used in the sense of Retakh, Serconek and Wilson. Graded is used in terms of Stanley's definition that all maximal chains have the same length n.

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

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 (9,-27,30,-9).

Cf. A206950 (maximal element removed).

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

Join[{0}, LinearRecurrence[{9, -27, 30, -9}, {0, 0, 3, 24}, 40]]

def a(n,adict={0:0,1:0,2:0,3:3,4:24}):
    if n in adict:
        return adict[n]
    adict[n]=9*a(n-1)-27*a(n-2)+30*a(n-3)-9*a(n-4)
    return adict[n]

a(n) = 9*a(n-1) - 27*a(n-2) + 30*a(n-3) - 9*a(n-4), a(1)=0, a(2)=0, a(3)=3, a(4)=24.
G.f.: (3*(1-x)*x^3)/((1-3*x)*(1-6*x+9*x^2-3*x^3)).
a(n) = A124292(n+1) - A025192(n).

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

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A206947 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank above 0.

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A206949 Number of nonisomorphic graded posets with 0 and non-uniform Hasse graph of rank n, with no 3-element antichain.

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Keywords

Comments

References

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Crossrefs

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Mathematica

Python

Formula