A339280
Number of unoriented series-parallel networks with n elements of 2 colors.
Original entry on oeis.org
2, 6, 22, 106, 582, 3622, 24060, 167803, 1206852, 8881775, 66496238, 504729590, 3874281594, 30020527274, 234498941338, 1844550287865, 14597849688004, 116151844649673, 928633009522942, 7456338969251761, 60101579662366508, 486145542528043029, 3944844113529346468
Offset: 1
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'.
a(1) = 2: (1), (2).
a(2) = 6: (11), (12), (22), (1|1), (1|2), (2|2).
A339283
Number of unoriented series-parallel networks with n integer valued elements spanning an initial interval of positive integers.
Original entry on oeis.org
1, 4, 28, 361, 7061, 184327, 5941855, 226973560, 10011116097, 500553647373, 27975194135702, 1728193768303770, 116934429186262096, 8600448307248025405, 683181845460624644202, 58290243136791332908001, 5316517137637684870655592, 516199318599186277653647746
Offset: 1
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'.
a(1) = 1: (1).
a(2) = 4: (11), (12), (1|1), (1|2).
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\\ See A339282 for R(n,k).
seq(n)={sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )}
A339281
Number of unoriented series-parallel networks with n colored elements using exactly 2 colors.
Original entry on oeis.org
0, 2, 14, 84, 522, 3426, 23404, 165417, 1197934, 8847201, 66359672, 504180138, 3872043674, 30011312118, 234460670790, 1844390161675, 14597175479270, 116148990100435, 928620864502940, 7456287071153017, 60101357023288316, 486144584042042269, 3944839973878931780
Offset: 1
In the following examples elements in series are juxtaposed and elements in parallel are separated by '|'.
a(2) = 2: (12), (1|2).
a(3) = 14: (112), (121), (122), (212), (1(1|2)), (1(2|2)), (2(1|1)), (2(1|2)), (1|12), (1|22), (2|11), (2|12), (1|1|2), (1|2|2).
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
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).
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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])) }
A339287
Number of inequivalent colorings of unoriented series-parallel networks with n colored elements.
Original entry on oeis.org
1, 4, 15, 105, 873, 9997, 134582, 2096206, 36391653, 693779666, 14346005530, 319042302578, 7579064231400, 191264021808301, 5103735168371201, 143438421861618397, 4231407420255210941, 130633362289335958866, 4209546674788934624394, 141259712052820378949746
Offset: 1
In the following examples elements in series are juxtaposed and elements in parallel are separated by '|'.
a(1) = 1: (1).
a(2) = 4: (11), (12), (1|1), (1|2).
a(3) = 15: (111), (112), (121), (123), (1(1|1)), (1(1|2)), (1(2|2)), (1(2|3)), (1|1|1), (1|1|2), (1|2|3), (1|11), (1|12), (1|22), (1|23).
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\\ See links in A339645 for combinatorial species functions.
B(n)={my(Z=x*sv(1), p=Z+O(x^2)); for(n=2, n, p=sEulerT(p^2/(1+p) + Z)-1); p}
cycleIndexSeries(n)={my(Z=x*sv(1), q=sRaise(B((n+1)\2), 2), s=x^2*sv(2)+q^2/(1+q), p=Z+O(x^2), t=p); for(n=1, n\2, t=Z + q*(1 + p); p=Z + sEulerT(t+(s-sRaise(t, 2))/2) - t - 1); (p+t-Z+B(n))/2}
InequivalentColoringsSeq(cycleIndexSeries(15))
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
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).
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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))}
Showing 1-6 of 6 results.
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