A339228
Triangle read by rows: T(n,k) is the number of oriented series-parallel networks with n colored elements using exactly k colors.
Original entry on oeis.org
1, 2, 3, 5, 22, 19, 15, 146, 321, 195, 48, 970, 4116, 5972, 2791, 167, 6601, 48245, 125778, 135235, 51303, 602, 46012, 546570, 2281528, 4238415, 3609966, 1152019, 2256, 328188, 6118320, 38437972, 109815445, 157612413, 111006329, 30564075
Offset: 1
Triangle begins:
1;
2, 3;
5, 22, 19;
15, 146, 321, 195;
48, 970, 4116, 5972, 2791;
167, 6601, 48245, 125778, 135235, 51303;
602, 46012, 546570, 2281528, 4238415, 3609966, 1152019;
...
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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)}
R(n,k)={my(Z=k*x, p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+Z)))); Vec(p)}
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]))}
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)) )}
A339230
Number of oriented series-parallel networks with integer valued elements summing to n.
Original entry on oeis.org
1, 3, 9, 32, 120, 490, 2077, 9158, 41401, 191232, 897849, 4273794, 20573696, 99994830, 490000756, 2418246995, 12008813611, 59962351145, 300864703306, 1516196518032, 7670827035223, 38946578808655, 198379559337073, 1013452414823740, 5191372465942866, 26658747310696437
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) = 9: (3), (12), (21), (1(1|1)), ((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)}
EdgeWeightedT(u)={my(Z=x*Ser(u)); my(p=Z+O(x^2)); for(n=2, #u, p=x*Ser(EulerT(Vec(p^2/(1+p)+Z)))); Vec(p)}
seq(n)={EdgeWeightedT(vector(n,i,1))}
A339233
Number of inequivalent colorings of oriented series-parallel networks with n colored elements.
Original entry on oeis.org
1, 4, 21, 165, 1609, 19236, 266251, 4175367, 72705802, 1387084926, 28689560868, 638068960017, 15158039092293, 382527449091778, 10207466648995608, 286876818184163613, 8462814670769394769, 261266723355912507073, 8419093340955799898258, 282519424041100564770142
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) = 21: (111), (112), (121), (122), (123), (1(1|1)), (1(1|2)), (1(2|2)), (1(2|3)), ((1|1)1), ((1|1)2), ((1|2)1), ((1|2)3), (1|1|1), (1|1|2), (1|2|3), (1|11), (1|12), (1|21), (1|22), (1|23).
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\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(Z=x*sv(1), p=Z+O(x^2)); for(n=2, n, p=sEulerT(p^2/(1+p) + Z)-1); p}
InequivalentColoringsSeq(cycleIndexSeries(15))
A339298
Number of oriented series-parallel networks with n integer valued elements spanning an initial interval of positive integers and without multiple unit elements in parallel.
Original entry on oeis.org
1, 3, 26, 339, 6241, 146261, 4192048, 141800728, 5532432894, 244530744314, 12076710992326, 659091388208810, 39390323152318533, 2558571976136883890, 179469998189344217166, 13520418589045722826274, 1088748555389998857776521, 93324804498600668393385498
Offset: 1
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\\ See A339297 for R(n, k).
seq(n)={sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )}
Showing 1-5 of 5 results.
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