A339156 -id:A339156 - 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

A339290 Number of oriented series-parallel networks with n elements and without multiple unit elements in parallel.

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 103, 306, 930, 2887, 9100, 29082, 93951, 306414, 1007361, 3335088, 11108986, 37203873, 125193694, 423099557, 1435427202, 4886975378, 16690971648, 57172387872, 196358421066, 676050576441, 2332887221847, 8067160995797, 27950871439353, 97019613539949

Offset: 1

Andrew Howroyd, Dec 07 2020

nonn

A series configuration is an ordered concatenation of two or more parallel configurations and a parallel configuration is a multiset of two or more unit elements or series configurations. In this variation, parallel configurations may include the unit element only once. 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) = 2: (ooo), (o|oo).
a(4) = 5: (oooo), (o(o|oo)), ((o|oo)o), (o|ooo), (oo|oo).
a(5) = 13: (ooooo), (oo(o|oo)), (o(o|oo)o), ((o|oo)oo), (o(o|ooo)), (o(oo|oo)), ((o|ooo)o), ((oo|oo)o), (o|oooo), (o|o(o|oo)), (o|(o|oo)o), (oo|ooo), (o|oo|oo).

Andrew Howroyd, Table of n, a(n) for n = 1..500

A003430 is the case with multiple unit elements in parallel allowed.
A058387 is the case that order is not significant in series configurations.
Cf. A339156, A339288, A339289, A339293 (achiral), A339296 (unoriented), A339301 (labeled).

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

a(n) = A339288(n) + A339289(n).

G.f.: P(x)/(1 - P(x)) where P(x) is the g.f. of A339289.

A339154 Number of essentially series oriented series-parallel networks with n elements and without unit elements in parallel.

Original entry on oeis.org

0, 1, 1, 1, 3, 6, 14, 30, 70, 165, 397, 961, 2368, 5875, 14722, 37134, 94312, 240823, 618147, 1593606, 4125218, 10717064, 27934867, 73032798, 191464677, 503218042, 1325678981, 3499913710, 9258627528, 24538328431, 65147600774, 173243773337, 461400769439

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 number of series 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(2) = 1: (oo).
a(3) = 1: (ooo).
a(4) = 1: (oooo).
a(5) = 3: (ooooo), (o(oo|oo)), ((oo|oo)o).
a(6) = 6: (oooooo), (oo(oo|oo)), (o(oo|oo)o), ((oo|oo)oo), (o(oo|ooo)), ((oo|ooo)o).

Cf. A003430, A339151, A339155, A339156.

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

G.f.: P(x)^2/(1 - P(x)) where P(x) is the g.f. of A339155.

G.f.: B(x)^2/(1 + B(x)) where B(x) is the g.f. of A339156.

A339155 Number of essentially parallel oriented series-parallel networks with n elements and without unit elements in parallel.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 5, 13, 29, 70, 165, 409, 1001, 2505, 6278, 15904, 40447, 103567, 266229, 687668, 1782573, 4637731, 12103112, 31679212, 83135973, 218713492, 576683119, 1523740365, 4033915677, 10698680606, 28422818782, 75629586540, 201539697208, 537818080714

Offset: 1

Andrew Howroyd, Nov 26 2020

nonn

			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(4) = 1: (oo|oo).
a(5) = 1: (oo|ooo).
a(6) = 3: (oo|oooo), (ooo|ooo), (oo|oo|oo).
a(7) = 4: (oo|ooooo), (oo|o(oo|oo)), (oo|(oo|oo)o), (ooo|oooo), (oo|oo|ooo).

Cf. A003430, A339152, A339154, A339156.

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(1 - 1/(1+p))}

G.f.: B(x)/(1 + B(x)) where B(x) is the g.f. of A339156.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A339290 Number of oriented series-parallel networks with n elements and without multiple unit elements in parallel.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A339154 Number of essentially series oriented series-parallel networks with n elements and without unit elements in parallel.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A339155 Number of essentially parallel oriented series-parallel networks with n elements and without unit elements in parallel.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula