A003430 -id:A003430 - cp's OEIS frontend

A202182 Number of n-element unlabeled N-free posets.

1, 2, 5, 15, 49, 180, 715, 3081, 14217, 69905, 363926, 1996922, 1150036, 69269925

Offset: 1

N. J. A. Sloane, Dec 13 2011

An N-free poset is a poset that does not contain any four elements x, y, z, w such that x is covered by z, y is covered by z, y is covered by w, x||y, x||w, and z||w. - Salah Uddin Mohammad, Mar 20 2021

			From _Salah Uddin Mohammad_, Mar 20 2021: (Start)
The following poset on 5 nodes is considered to be N-free in this sequence.
     o     o
     | \   |
     |  o  |
     |   \ |
     o     o
An alternative definition of N-free used by A003430 excludes this.
(End)

Soheir M. Khamis, Height counting of unlabeled interval and N-free posets, Discrete Math. 275 (2004), no. 1-3, 165-175.
Salah Uddin Mohammad, Md. Shah Noor, and Md. Rashed Talukder, An Exact Enumeration of the Unlabeled Disconnected Posets, J. Int. Seq., Vol. 25 (2022), Article 22.5.4.

Row sums of A202181.

Cf. A003430.

Missing a(12) inserted by Salah Uddin Mohammad, Mar 20 2021

A331156 Number of (weakly) connected gluing-parallel or GP-posets with n points.

Original entry on oeis.org

1, 1, 1, 3, 10, 44, 233

Offset: 0

N. J. A. Sloane, Jan 16 2020, following a suggestion from Michael De Vlieger

Uli Fahrenberg, Christian Johansen, Georg Struth, Ratan Bahadur Thapa, Generating Posets Beyond N, arXiv:1910.06162 [cs.FL], 2019.

The seven sequences in the table of Uli Fahrenberg et al., 2019, are A000112, A003430, A079566, A331156, A331157, A331158, A331159.

Typo in a(6) corrected by Uli Fahrenberg, Feb 03 2024

A331157 Number of iposets (posets with interfaces) with starting interfaces only, with n points.

Original entry on oeis.org

1, 2, 5, 16, 66, 350

Offset: 0

N. J. A. Sloane, Jan 16 2020, following a suggestion from Michael De Vlieger

Uli Fahrenberg, Christian Johansen, Georg Struth, Ratan Bahadur Thapa, Generating Posets Beyond N, arXiv:1910.06162 [cs.FL], 2019.

The seven sequences in the table of Uli Fahrenberg et al., 2019, are A000112, A003430, A079566, A331156, A331157, A331158, A331159.

A331158 Number of iposets (posets with interfaces) with n points.

Original entry on oeis.org

1, 4, 17, 86, 532, 4068, 38933, 474822, 7558620

Offset: 0

N. J. A. Sloane, Jan 16 2020, following a suggestion from Michael De Vlieger

Uli Fahrenberg, Christian Johansen, Georg Struth, Ratan Bahadur Thapa, Generating Posets Beyond N, arXiv:1910.06162 [cs.FL], 2019.
Uli Fahrenberg, Christian Johansen, Georg Struth, Krzysztof Ziemiański, Posets with Interfaces for Concurrent Kleene Algebra, arxiv:2106.10895 [cs.FL], 2021.

The seven sequences in the table of Uli Fahrenberg et al., 2019, are A000112, A003430, A079566, A331156, A331157, A331158, A331159.

a(5)-a(8) from Uli Fahrenberg, Jun 22 2021

A331159 Number of GP-iposets (gluing-parallel posets with interfaces) with n points.

Original entry on oeis.org

1, 4, 16, 74, 419, 2980, 26566, 289279, 3726311

Offset: 0

N. J. A. Sloane, Jan 16 2020, following a suggestion from Michael De Vlieger

Uli Fahrenberg, Christian Johansen, Georg Struth, Ratan Bahadur Thapa, Generating Posets Beyond N, arXiv:1910.06162 [cs.FL], 2019.
Uli Fahrenberg, Christian Johansen, Georg Struth, Krzysztof Ziemiański, Posets with Interfaces for Concurrent Kleene Algebra, arxiv:2106.10895 [cs.FL], 2021.

The seven sequences in the table of Uli Fahrenberg et al., 2019, are A000112, A003430, A079566, A331156, A331157, A331158, A331159.

a(7)-a(8) from Uli Fahrenberg, Jun 22 2021

A339156 Number of oriented series-parallel networks with n elements and without unit elements in parallel.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 19, 43, 99, 235, 562, 1370, 3369, 8380, 21000, 53038, 134759, 344390, 884376, 2281274, 5907791, 15354795, 40037979, 104712010, 274600650, 721931534, 1902362100, 5023654075, 13292543205, 35237009037, 93570419556, 248873359877, 662940466647

Offset: 1

Andrew Howroyd, Nov 26 2020

nonn

A series configuration is an ordered concatenation of two or more parallel configurations and a parallel configuration is the unit element or a multiset of two or more series configurations. a(n) is the total number of series and parallel configurations with n unit elements.

			In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(1) = 1: (o)
a(2) = 1: (oo).
a(3) = 1: (ooo).
a(4) = 2: (oooo), (oo|oo).
a(5) = 4: (ooooo), (o(oo|oo)), ((oo|oo)o), (oo|ooo).
a(6) = 9: (oooooo), (oo(oo|oo)), (o(oo|oo)o), ((oo|oo)oo), (o(oo|ooo)), ((oo|ooo)o), (oo|oooo), (ooo|ooo), (oo|oo).

Cf. A003430, A339153, A339154, A339155.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=x+O(x^2)); for(n=2, n, p=x+x*Ser(EulerT(Vec(p^2/(1+p), -n)))); Vec(p)}

G.f.: A(x) where A(x) satisfies A(x) = x - 1 + exp(Sum_{k>=1} (A(x^k) + 1/(1 + A(x^k)) - 1)/k).
a(n) = A339154(n) + A339155(n).
Euler transform of A339154 gives this sequence with a(1) = 0.
G.f.: P(x)/(1 - P(x)) where P(x) is the g.f. of A339155.
G.f.: S(x)/2 + sqrt(S(x) + S(x)^2/4) where S(x) is the g.f. of A339154.

A339158 Number of essentially parallel achiral series-parallel networks with n elements.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 37, 84, 180, 413, 902, 2084, 4628, 10726, 24128, 56085, 127421, 296955, 680092, 1588665, 3662439, 8574262, 19875081, 46628789, 108584460, 255264307, 596774173, 1405626896, 3297314994, 7780687159, 18305763571, 43271547808, 102069399803

Offset: 1

Andrew Howroyd, Nov 27 2020

nonn

A series configuration is the unit element or an ordered concatenation of two or more parallel configurations and a parallel configuration is the unit element or a multiset of two or more series configurations. a(n) is the number of parallel configurations with n unit elements that are invariant under the reversal of all contained series configurations.

			In the following examples of series-parallel networks, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (o|o).
a(3) = 2: (o|oo).
a(4) = 4: (o|ooo), (oo|oo), (o|o|oo), (o|o|o|o).
a(5) = 8: (o|oooo), (o|(o|o)(o|o)), (o|o(o|o)o), (oo|ooo), (o|o|ooo), (o|oo|oo), (o|o|o|oo), (o|o|o|o|o).
a(6) = 16 includes (o(o|o)|(o|o)o) which is the first example of a network that is achiral but does not have reflective symmetry when embedded in the plane as shown below (edges correspond to elements):
               A
             /   \\
            o      o   --- No reflective symmetry ---
             \\  /
               Z

Cf. A003430, A007454 (oriented), A339157, A339159, A339224 (unoriented).

\\ here B(n) gives A003430 as a power series.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
B(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x)))); p}
seq(n)={my(q=subst(B((n+1)\2), x, x^2), s=x^2+q^2/(1+q), p=x+O(x^2)); for(n=1, n\2, my(t=x + q*(1 + p)); p=x + x*Ser(EulerT(Vec(t+(s-subst(t,x,x^2))/2))) - t); Vec(p+O(x*x^n))}

G.f.: x - S(x) - 1 + exp(Sum_{k>=1} (S(x^k) + (R(x^(2*k)) - S(x^(2*k)))/2)/k) where S(x) is the g.f. of A339157 and R(x) is the g.f. of A007453.

A339224 Number of essentially parallel unoriented series-parallel networks with n elements.

Original entry on oeis.org

1, 1, 2, 5, 13, 41, 132, 470, 1730, 6649, 26122, 104814, 426257, 1754055, 7282630, 30470129, 128304158, 543303752, 2311904374, 9880776407, 42394198909, 182537610058, 788473887942, 3415782381520, 14837307126498, 64608442956047, 281975101347994, 1233237605651194

Offset: 1

Andrew Howroyd, Nov 27 2020

nonn

See A339225 for additional details.

			In the following examples of series-parallel networks, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (oo), (o|o).
a(3) = 2: (o|o|o), (o|oo).
a(4) = 5: (o|o|o|o), (o|o|oo), (oo|oo), (o|ooo), (o|o(o|o)).

Cf. A003430, A007454 (oriented), A339158 (achiral), A339223, A339225.

\\ here B(n) gives A003430 as a power series.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
B(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x)))); p}
seq(n)={my(q=subst(B((n+1)\2), x, x^2), s=x^2+q^2/(1+q), p=x+O(x^2)); for(n=1, n\2, my(t=x + q*(1 + p)); p=x + x*Ser(EulerT(Vec(t+(s-subst(t, x, x^2))/2))) - t); Vec(p+subst(x/(1+x), x, B(n)))/2}

a(n) = (A007454(n) + A339158(n))/2.

A339226 Number of oriented series-parallel networks with n elements of 2 colors.

Original entry on oeis.org

2, 7, 32, 176, 1066, 6935, 47216, 332700, 2404818, 17734668, 132901644, 1009161505, 7747608480, 60037905076, 468987635982, 3689066578347, 29195587558726, 232303316402615, 1857264782113562, 14912673794505898, 120203145484455930, 972291038495626309

Offset: 1

Andrew Howroyd, Nov 28 2020

nonn

See A339228 for additional details.

			In the following examples elements in series are juxtaposed and elements in parallel are separated by '|'.
a(1) = 2: (1), (2).
a(2) = 7: (11), (12), (21), (22), (1|1), (1|2), (2|2).

Cf. A003430, A339227, A339228.

\\ See A339228 for R(n,k).
seq(n) = {R(n,2)}

A349276 Number of unlabeled P-series with n elements.

Original entry on oeis.org

1, 2, 5, 13, 31, 76, 178, 423, 988, 2312, 5361, 12427, 28626, 65813, 150700, 344232, 783832, 1780650, 4034591, 9121571, 20576349, 46322816, 104079338, 233421517, 522574991, 1167974002, 2606282841, 5806953923, 12919314397, 28702716868, 63682839588, 141111193270

Offset: 1

Salah Uddin Mohammad, Nov 12 2021

nonn

The class of all P-series is a subclass of the class of series-parallel posets and it contains the class of P-graphs as a subclass.
A poset is called a P-graph if it can be expressed as the ordinal sum of the antichain posets (including the singleton poset).
A poset is called a P-series if it is either a P-graph or it can be expressed as the direct sum of the P-graphs.
For example, all the 3-element posets are P-series, where only the connected posets and the antichains are P-graphs. On the other hand, the 4-element poset <{x,y,z,w},{x<.z, z<.w, y<.w, x||y, y||z}> and its dual are both series-parallel which are not the P-series. Here, by 'x<.z' we mean 'x is covered by z'.

Alois P. Heinz, Table of n, a(n) for n = 1..3217

Cf. A003430 (series-parallel posets), A255047, A349488.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
      max(1, 2^(d-1)-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 05 2022

a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j]*Sum[d*
     Max[1, 2^(d - 1) - 1], {d, Divisors[j]}], {j, 1, n}]/n];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={EulerT(Vec((1 -2*x +2*x^2)/((1-x)*(1-2*x)) + O(x*x^n)))} \\ Andrew Howroyd, Nov 19 2021

a(n) = A255047(n-1) + A349488(n).

G.f: -1 + exp(Sum_{k>=1} B(x^k)/k) where B(x) = x*(1 - 2*x + 2*x^2)/((1 - x)*(1 - 2*x)). - Andrew Howroyd, Jan 06 2022

cp's OEIS Frontend

A202182 Number of n-element unlabeled N-free posets.

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A331156 Number of (weakly) connected gluing-parallel or GP-posets with n points.

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A331157 Number of iposets (posets with interfaces) with starting interfaces only, with n points.

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A331158 Number of iposets (posets with interfaces) with n points.

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A331159 Number of GP-iposets (gluing-parallel posets with interfaces) with n points.

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A339156 Number of oriented series-parallel networks with n elements and without unit elements in parallel.

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A339158 Number of essentially parallel achiral series-parallel networks with n elements.

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A339224 Number of essentially parallel unoriented series-parallel networks with n elements.

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A339226 Number of oriented series-parallel networks with n elements of 2 colors.

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PARI

A349276 Number of unlabeled P-series with n elements.

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