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]))}
A339227
Number of oriented series-parallel networks with n colored elements using exactly 2 colors.
Original entry on oeis.org
0, 3, 22, 146, 970, 6601, 46012, 328188, 2387498, 17666752, 132631060, 1008068661, 7743145556, 60019505338, 468911161556, 3688746483355, 29194239490432, 232297608127077, 1857240493924050, 14912570002666430, 120202700216204324, 972289121546949231
Offset: 1
In the following examples elements in series are juxtaposed and elements in parallel are separated by '|'.
a(2) = 3: (12), (21), (22), (1|2).
a(3) = 22: (112), (121), (122), (211), (212), (221), (1(1|2)), (1(2|2)), (2(1|1)), (2(1|2)), ((1|1)2), ((1|2)1), ((1|2)2), ((2|2)1), (1|12), (1|21), (1|22), (2|21), (11|2), (12|2), (1|1|2), (1|2|2).
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).
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))
Showing 1-4 of 4 results.
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