A339282
Triangle read by rows: T(n,k) is the number of unoriented series-parallel networks with n colored elements using exactly k colors.
Original entry on oeis.org
1, 2, 2, 4, 14, 10, 11, 84, 168, 98, 30, 522, 2109, 3004, 1396, 98, 3426, 24397, 63094, 67660, 25652, 328, 23404, 274626, 1142420, 2119985, 1805082, 576010, 1193, 165417, 3065376, 19230320, 54916745, 78809079, 55503392, 15282038, 4459, 1197934, 34201068, 311157620, 1283360335, 2761083930, 3220245007, 1932118328, 467747416
Offset: 1
Triangle begins:
1;
2, 2;
4, 14, 10;
11, 84, 168, 98;
30, 522, 2109, 3004, 1396;
98, 3426, 24397, 63094, 67660, 25652;
328, 23404, 274626, 1142420, 2119985, 1805082, 576010;
...
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\\ R(n, k) gives colorings using at most k colors as a vector.
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}
R(n,k)={my(Z=k*x, 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}
M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))}
{my(T=M(8)); for(n=1, #T~, print(T[n, 1..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 1-3 of 3 results.
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