A339230 -id:A339230 - cp's OEIS frontend

A339231 Triangle read by rows: T(n,k) is the number of oriented series-parallel networks whose multigraph has n edges and k interior vertices, 0 <= k < n.

1, 1, 1, 1, 3, 1, 1, 6, 7, 1, 1, 10, 23, 13, 1, 1, 15, 59, 69, 22, 1, 1, 21, 124, 249, 172, 34, 1, 1, 28, 234, 711, 853, 378, 50, 1, 1, 36, 402, 1733, 3175, 2487, 755, 70, 1, 1, 45, 650, 3755, 9767, 11813, 6431, 1400, 95, 1, 1, 55, 995, 7443, 26043, 44926, 38160, 15098, 2445, 125, 1

Offset: 1

Andrew Howroyd, Nov 29 2020

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. T(n, k) is the number of series or parallel configurations with n unit elements whose representation as a multigraph has k interior vertices, with elements corresponding to edges. Parallel configurations do not increase the interior vertex count and series configurations increase it by one less than the number of parts.

			Triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   7,   1;
  1, 10,  23,  13,   1;
  1, 15,  59,  69,  22,   1;
  1, 21, 124, 249, 172,  34,  1;
  1, 28, 234, 711, 853, 378, 50, 1;
  ...
In the following examples elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
T(4,0) = 1: (o|o|o|o).
T(4,1) = 6: ((o|o)(o|o)), (o(o|o|o)), ((o|o|o)o), (o|o|oo), (o|o(o|o)), (o|(o|o)o).
T(4,2) = 7: (oo(o|o)), (o(o|o)o), ((o|o)oo),  (o(o|oo)), ((o|oo)o),  (oo|oo), (o|ooo).
T(4,3) = 1: (oooo).
The graph of (oo(o|o)) has 4 edges (elements) and 2 interior vertices as shown below:
      A---o---o===Z (where === is a double edge).

Row sums are A003430.

Cf. A002623, A339228, A339230, A339232.

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)}
VertexWeighted(n, W)={my(Z=x, p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerMT(Vec(W*p^2/(1 + W*p) + Z)))); Vec(p)}
T(n)={[Vecrev(p)|p<-VertexWeighted(n,y)]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) }

T(n,0) = T(n,n-1) = 1.
T(n,1) = binomial(n,2).
T(n+2,n) = A002623(n).
Sum_{k=1..n-1} k*T(n,k) = A339232(n).

A339284 Number of unoriented series-parallel networks with integer valued elements summing to n.

Original entry on oeis.org

1, 3, 7, 23, 73, 281, 1112, 4779, 21139, 96793, 451631, 2144101, 10303984, 50042734, 245110900, 1209414659, 6005130171, 29983077169, 150437143336, 758110844897, 3835445581758, 19473373629628, 99189996107004, 506726776334889, 2595687705113097

Offset: 1

Andrew Howroyd, Nov 30 2020

nonn

See A339282 for additional details.

			In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'.
a(1) = 1: (1).
a(2) = 3: (2), (11), (1|1).
a(3) = 7: (3), (12), (1(1|1)), (111), (1|2), (1|11), (1|1|1).

Cf. A339225, A339230, A339282, A339283.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
B(n, Z)={my(p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+Z)))); p}
EdgeWeightedT(u)={my(Z=x*Ser(u), n=#u, q=subst(B((n+1)\2, Z), x, x^2), s=subst(Z,x,x^2)+q^2/(1+q), p=Z+O(x^2), t=p); for(n=1, n\2, t=Z + q*(1 + p); p=Z + x*Ser(EulerT(Vec(t+(s-subst(t, x, x^2))/2))) - t); Vec(p+t-Z+B(n,Z))/2}
seq(n)={EdgeWeightedT(vector(n,i,1))}

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A339231 Triangle read by rows: T(n,k) is the number of oriented series-parallel networks whose multigraph has n edges and k interior vertices, 0 <= k < n.

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