A206901 -id:A206901 - 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)).

A206902 Number of nonisomorphic graded posets with 0 and uniform Hasse diagram of rank n with no 3-element antichain.

Original entry on oeis.org

1, 2, 8, 36, 166, 768, 3554, 16446, 76102, 352152, 1629536, 7540458, 34892452, 161460114, 747134894, 3457265922, 15998031616, 74028732924, 342557973998, 1585140808368, 7335025230994, 33941839649382, 157061283704438, 726779900373936, 3363075935260696

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.

Uniform (in the definition) used in the sense of Retakh, Serconek and Wilson (see paper in Links lines). - David Nacin, Mar 01 2012

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 (6,-7,3).

Cf. A025192 (adding a unique maximal element).

Cf. A124292, A206901 (dropping uniformity with and without maximal element).

a:=[2,8,36];; for n in [4..30] do a[n]:=6*a[n-1]-7*a[n-2]+3*a[n-3]; od; Concatenation([1], a); # G. C. Greubel, May 21 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-4*x +3*x^2-x^3)/(1-6*x+7*x^2-3*x^3) )); // G. C. Greubel, May 21 2019

LinearRecurrence[{6,-7,3}, {1,2,8,36}, 30] (* Vincenzo Librandi, Feb 27 2012 *)

my(x='x+O('x^30)); Vec((1-4*x+3*x^2-x^3)/(1-6*x+7*x^2-3*x^3)) \\ G. C. Greubel, May 21 2019

def a(n, adict={1:2,2:8,3:36}):
    if n in adict:
        return adict[n]
    adict[n]=6*a(n-1)-7*a(n-2)+3*a(n-3)
    return adict[n]

((1-4*x+3*x^2-x^3)/(1-6*x+7*x^2-3*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 21 2019

a(n) = 6*a(n-1) - 7*a(n-2) + 3*a(n-3), a(1)=2, a(2)=8, a(3)=36.

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

A208736 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level between 0 and 1.

Original entry on oeis.org

0, 0, 0, 1, 5, 22, 91, 361, 1392, 5265, 19653, 72694, 267179, 977593, 3565600, 12975457, 47142021, 171075606, 620303547, 2247803785, 8141857808, 29481675889, 106728951109, 386314552438, 1398132674955, 5059626441177, 18308871648576, 66249898660801

Offset: 0

David Nacin, Mar 01 2012

Uniform used in the sense of Retakh, Serconek and Wilson. We use Stanley's definition of graded poset: all maximal chains have the same length n (which also implies all maximal elements have maximal rank.)

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

Cf. A208737, A206901, A206902, A206947-A206950, A001906, A025192, A081567, A124302, A124292, A088305, A086405, A012781.

Join[{0, 0}, LinearRecurrence[{8, -21, 20, -5}, {0, 1, 5, 22}, 40]]

def a(n, d={0:0,1:0,2:0,3:1,4:5,5:22}):
    if n in d:
        return d[n]
    d[n]=8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4)
    return d[n]

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), a(2) = 0, a(3) = 1, a(4) = 5, a(5) = 22.
G.f.: (x^3 - 3*x^4 + 3*x^5)/(1 - 8*x + 21*x^2 - 20*x^3 + 5*x^4); (x^3 * (1 - 3*x + 3*x^2))/((1 - 3*x + x^2)*(1 - 5*x + 5*x^2)) .
a(n) = A081567(n-2) - A001519(n-1).

A208737 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with no 3-element antichain.

Original entry on oeis.org

0, 0, 0, 1, 7, 37, 175, 778, 3325, 13837, 56524, 227866, 909832, 3607294, 14227447, 55894252, 218937532, 855650749, 3338323915, 13007422705, 50631143323, 196928737582, 765495534433, 2974251390529, 11552064922624, 44856304154086

Offset: 0

David Nacin, Mar 01 2012

Uniform used in the sense of Retakh, Serconek and Wilson. We use Stanley's definition of graded poset: all maximal chains have the same length n (which also implies all maximal elements have maximal rank.)

R. 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 (10,-36,57,-39,9).

Cf. A208736, A206901, A206902, A206947-A206950, A001906, A025192, A081567, A124302, A124292, A088305, A086405, A012781.

Join[{0}, LinearRecurrence[{10, -36, 57, -39, 9}, {0, 0, 1, 7, 37}, 40]]

def a(n, d={0:0,1:0,2:0,3:1,4:7,5:37}):
    if n in d:
        return d[n]
    d[n]=10*a(n-1) - 36*a(n-2) + 57*a(n-3) - 39*a(n-4) + 9*a(n-5)
    return d[n]

a(n) = 10*a(n-1) - 36*a(n-2) + 57*a(n-3) - 39*a(n-4) + 9*a(n-5), a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 7, a(5) = 37.
G.f: (x^3 - 3*x^4 + 3*x^5)/(1 - 10*x + 36*x^2 - 57*x^3 + 39*x^4 - 9*x^5); (x^3*(1 - 3*x + 3*x^2)) / ((1 - x) (1 - 3*x) (1 - 6*x + 9*x^2 - 3*x^3)).
a(n) = A124292(n) - A124302(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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A206902 Number of nonisomorphic graded posets with 0 and uniform Hasse diagram of rank n with no 3-element antichain.

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A208736 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level between 0 and 1.

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A208737 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with no 3-element antichain.

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