A339231 -id:A339231 - 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

A339232 Total number of interior vertices in the multigraphs of all oriented series-parallel networks with n edges.

Original entry on oeis.org

0, 1, 5, 23, 99, 433, 1880, 8238, 36202, 159898, 708517, 3150128, 14042620, 62751693, 280997846, 1260635337, 5664870696, 25493707908, 114882350739, 518318733052, 2341079272919, 10584488664085, 47898510357544, 216940538748652, 983326680302665, 4460343301915203

Offset: 1

Andrew Howroyd, Nov 29 2020

nonn

See A339231 for additional details.

Cf. A003430, A339231.

\\ See A339231 for VertexWeighted.
seq(n)={subst(deriv(VertexWeighted(n,y)), y, 1)}

a(n) = Sum_{k=1..n-1} k*A339231(n,k).

A339285 Triangle read by rows: T(n,k) is the number of unoriented series-parallel networks whose multigraph has n edges and k interior vertices, 0 <= k < n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 6, 14, 8, 1, 1, 9, 34, 39, 14, 1, 1, 12, 68, 132, 94, 20, 1, 1, 16, 126, 370, 447, 202, 30, 1, 1, 20, 212, 887, 1625, 1275, 398, 40, 1, 1, 25, 340, 1911, 4955, 5985, 3284, 730, 55, 1, 1, 30, 515, 3765, 13133, 22608, 19245, 7649, 1266, 70, 1

Offset: 1

Andrew Howroyd, Nov 30 2020

Unoriented version of A339231. Equivalence is up to reversal of all parts combined in series.

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   5,    1;
  1,  6,  14,    8,    1;
  1,  9,  34,   39,   14,    1;
  1, 12,  68,  132,   94,   20,    1;
  1, 16, 126,  370,  447,  202,   30,   1;
  1, 20, 212,  887, 1625, 1275,  398,  40,  1;
  1, 25, 340, 1911, 4955, 5985, 3284, 730, 55, 1;
  ...
T(4,0) = 1: (o|o|o|o).
T(4,1) = 4: ((o|o)(o|o)), (o(o|o|o)), (o|o|oo), (o|o(o|o)).
T(4,2) = 5: (oo(o|o)), (o(o|o)o),  (o(o|oo)),  (oo|oo), (o|ooo).
T(4,3) = 1: (oooo).

Row sums are A339225.

Cf. A002620, A339231, A339282, A339286.

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, [v^i|v<-vars])/i ))-1)}
SubPwr(p,e)={my(vars=variables(p)); substvec(p, vars, [v^e|v<-vars])}
BW(n, Z, W)={my(p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerMT(Vec(W*p^2/(1+W*p)+Z)))); p}
VertexWeighted(n, Z, W)={my(q=SubPwr(BW((n+1)\2, Z, W), 2), W2=SubPwr(W, 2), s=SubPwr(Z, 2)+W2*q^2/(1+W2*q), p=Z+O(x^2), t=p); for(n=1, n\2, t=Z + q*(W + W2*p); p=Z + x*Ser(EulerMT(Vec(t+(s-SubPwr(t, 2))/2))) - t); Vec(p+t-Z+BW(n, Z, W))/2}
T(n)={[Vecrev(p)|p<-VertexWeighted(n, x, y)]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) }

T(n,0) = T(n,n-1) = 1.
T(n,1) = A002620(n).
A339286(n) = Sum_{k=1..n-1} k*T(n,k).

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

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A339232 Total number of interior vertices in the multigraphs of all oriented series-parallel networks with n edges.

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A339285 Triangle read by rows: T(n,k) is the number of unoriented series-parallel networks whose multigraph has n edges and k interior vertices, 0 <= k < n.

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