A003431 -id:A003431 - cp's OEIS frontend

A003430 Number of unlabeled series-parallel posets (i.e., generated by unions and sums) with n nodes.

1, 1, 2, 5, 15, 48, 167, 602, 2256, 8660, 33958, 135292, 546422, 2231462, 9199869, 38237213, 160047496, 674034147, 2854137769, 12144094756, 51895919734, 222634125803, 958474338539, 4139623680861, 17931324678301, 77880642231286, 339093495674090, 1479789701661116

Offset: 0

N. J. A. Sloane, Richard Stanley, and Mira Bernstein

Number of oriented series-parallel networks with n elements. A series configuration is a unit element or an ordered concatenation of two or more parallel configurations and a parallel configuration is a unit element or a multiset of two or more series configurations. a(n) is the number of series or parallel configurations with n elements. The sequences A007453 and A007454 enumerate respectively series and parallel configurations. - Andrew Howroyd, Dec 01 2020

			From _Andrew Howroyd_, Nov 26 2020: (Start)
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) = 2: (oo), (o|o).
a(3) = 5: (ooo), (o(o|o)), ((o|o)o), (o|o|o), (o|oo).
a(4) = 15: (oooo), (oo(o|o)), (o(o|o)o), ((o|o)oo), ((o|o)(o|o)), (o(o|oo)), (o(o|o|o)), ((o|oo)o), ((o|o|o)o), (o|o|o|o), (o|o|oo), (oo|oo), (o|ooo), (o|o(o|o)), (o|(o|o)o).
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39 (which deals with the labeled case of the same sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 1..100 from Jean-François Alcover)
B. I. Bayoumi, M. H. El-Zahar and S. M. Khamis, Asymptotic enumeration of N-free partial orders, Order 6 (1989), 219-232.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Uli Fahrenberg, Christian Johansen, Georg Struth, Ratan Bahadur Thapa, Generating Posets Beyond N, arXiv:1910.06162 [cs.FL], 2019.
Frédéric Fauvet, L. Foissy, D. Manchon, Operads of finite posets, arXiv preprint arXiv:1604.08149 [math.CO], 2016.
S. R. Finch, Series-parallel networks
S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 72.
Soheir M. Khamis, Height counting of unlabeled interval and N-free posets, Discrete Math. 275 (2004), no. 1-3, 165-175.
R. P. Stanley, Enumeration of posets generated by disjoint unions and ordinal sums, Proc. Amer. Math. Soc. 45 (1974), 295-299. Math. Rev. 50 #4416.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
Index entries for sequences related to posets

Row sums of A339231.
Column k=1 of A339228.
Cf. A000084, A003431, A048172 (labeled N-free posets), A007453, A007454, A339156, A339159, A339225.

terms = 25; A[] = 1; Do[A[x] = Exp[Sum[(1/k)*(A[x^k] + 1/A[x^k] - 2 + x^k), {k, 1, terms + 1}]] + O[x]^(terms + 1) // Normal, terms + 1];
CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jun 29 2011, updated Jan 12 2018 *)

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*Ser(EulerT(Vec(p^2/(1+p)+x, 1-n)))); Vec(p)} \\ Andrew Howroyd, Nov 27 2020

G.f. A(x) = 1 + x + 2*x^2 + 5*x^3 + ... satisfies A(x) = exp(Sum_{k>=1} (1/k)*(A(x^k) + 1/A(x^k) - 2 + x^k)).
From: Andrew Howroyd, Nov 26 2020: (Start)
a(n) = A007453(n) + A007454(n) for n > 1.
Euler transform of A007453.
G.f.: P(x)/(1 - P(x)) where P(x) is the g.f. of A007454.
(End)

Name corrected by Salah Uddin Mohammad, Jun 07 2020

a(0)=1 prepended (using the g.f.) by Alois P. Heinz, Dec 01 2020

A046906 Number of connected irreducible posets with n labeled points.

Original entry on oeis.org

1, 1, 0, 0, 24, 1080, 52440, 3281880, 277953144, 32418855000, 5239070305080, 1173944480658840, 363936227764858584, 155521768202208047640, 91218870039317505477720, 73113879800794757415243480, 79743817918540500914682249144, 117883366412734188786535902826200, 235329353612778837110901775412557560

Offset: 0

John A. Wright

nonn

J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.

S. R. Finch, Series-parallel networks
S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for sequences related to posets

A003431 gives isomorphism classes of these posets.

nn = 18; A[x_] := Total[Cases[Import["https://oeis.org/A001035/b001035.txt",
      "Table"], {, }][[All, 2]]*Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]];
Range[0, nn]! CoefficientList[ Series[(1 + Log[A[x]]) - A[ x] (1 - 1/A[x])^2 , {x, 0, nn}], x] (* Geoffrey Critzer, Jul 09 2022 *)

From Geoffrey Critzer, Jul 09 2022: (Start)
E.g.f.: 1 + log(A(x)) - A(x)(1-1/A(x))^2  where A(x) is the e.g.f. for A001035.
a(n) = A001927(n) - Sum_{k>=2} A354615(n,k). (End)

a(8)-a(18) from Geoffrey Critzer, Jul 09 2022

a(0) changed to 1 by Geoffrey Critzer, Jul 10 2022

cp's OEIS Frontend

A003430 Number of unlabeled series-parallel posets (i.e., generated by unions and sums) with n nodes.

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A046904 Number of isomorphism classes of posets with n points with property that there is no nonsingelton proper subset T for which x not in T implies xT or x incomparable with every element of T.

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A046905 Posets with n points with property that there is no nonsingelton proper subset T for which x not in T implies xT or x incomparable with every element of T.

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A046906 Number of connected irreducible posets with n labeled points.

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