A316977
Number of series-reduced rooted trees whose leaves are {1, 1, 2, 2, 3, 3, ..., n, n}.
Original entry on oeis.org
1, 12, 575, 66080, 13830706, 4566898564, 2181901435364, 1422774451251512, 1213875872220833664, 1312273759143855989808, 1752860078230602866012288, 2834766624822130489716563008, 5458358420687156358967526721408, 12339106957086349462329140342122112
Offset: 1
The a(2) = 12 trees are (1(1(22))), (1(2(12))), (1(122)), (2(1(12))), (2(2(11))), (2(112)), ((11)(22)), ((12)(12)), (11(22)), (12(12)), (22(11)), (1122).
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sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Length[gro[Ceiling[Range[1/2,n,1/2]]]],{n,4}]
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\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(2*n), vars=vector(2*n-2,i,sv(2+i))); v[1]=sv(1); for(n=2, #v, v[n] = substvec(polcoef( sExp(x*Ser(v[1..n])), n ), vars[1..n-2], vector(n-2))); sCartProd(x*Ser(v), 1/(1-x^2*symGroupCycleIndex(2)) + O(x*x^(2*n)))}
seq(n)={my(p=substvec(cycleIndexSeries(n), [sv(1), sv(2)], [1,1])); vector(n, n, polcoef(p,2*n))} \\ Andrew Howroyd, Jan 02 2021
A323389
The number of connected, unlabeled, undirected, edge-signed cubic graphs (admitting loops and multiedges) on 2n vertices where the degree of the first sign is 2 at each node.
Original entry on oeis.org
1, 2, 5, 19, 88, 553, 4619, 49137, 646815, 10053183, 178725865, 3555840644, 78048875298, 1871066903575, 48617053973267, 1360733669185473, 40810827325698897, 1305690378666580997, 44387116312631271929, 1597768080980647428027, 60710507893875818581964
Offset: 0
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\\ See A339645 for combinatorial species functions.
cycleIndexSeries(n)={1+sLog(sCartProd(sExp(dihedralGroupSeries(n)), sExp(symGroupCycleIndex(2)*x^2 + O(x*x^n))))}
seq(n)={Vec(substpol(OgfSeries(cycleIndexSeries(2*n)), x^2, x))} \\ Andrew Howroyd, May 05 2023
A323786
Number of non-isomorphic weight-n multisets of multisets of non-singleton multisets.
Original entry on oeis.org
1, 0, 2, 3, 19, 39, 200, 615, 2849, 11174, 52377, 239269, 1191090, 6041975, 32275288, 177797719, 1017833092, 6014562272, 36717301665, 230947360981, 1495562098099, 9956230757240, 68070158777759, 477439197541792, 3432259679880648, 25267209686664449
Offset: 0
Non-isomorphic representatives of the a(4) = 19 multiset partitions:
{{1111}} {{1112}} {{1123}} {{1234}}
{{11}{11}} {{1122}} {{11}{23}} {{12}{34}}
{{11}}{{11}} {{11}{12}} {{12}{13}} {{12}}{{34}}
{{11}{22}} {{11}}{{23}}
{{12}{12}} {{12}}{{13}}
{{11}}{{12}}
{{11}}{{22}}
{{12}}{{12}}
Non-isomorphic representatives of the a(5) = 39 multiset partitions:
{{11111}} {{11112}} {{11123}} {{11234}} {{12345}}
{{11}{111}} {{11122}} {{11223}} {{11}{234}} {{12}{345}}
{{11}}{{111}} {{11}{112}} {{11}{123}} {{12}{134}} {{12}}{{345}}
{{11}{122}} {{11}{223}} {{23}{114}}
{{12}{111}} {{12}{113}} {{11}}{{234}}
{{12}{112}} {{12}{123}} {{12}}{{134}}
{{22}{111}} {{13}{122}} {{23}}{{114}}
{{11}}{{112}} {{23}{111}}
{{11}}{{122}} {{11}}{{123}}
{{12}}{{111}} {{11}}{{223}}
{{12}}{{112}} {{12}}{{113}}
{{22}}{{111}} {{12}}{{123}}
{{13}}{{122}}
{{23}}{{111}}
A339234
Number of series-reduced tanglegrams with n unlabeled leaves.
Original entry on oeis.org
1, 1, 5, 51, 757, 16416, 461231, 16021550, 662197510, 31749450007, 1732478051823, 106025572201434, 7192665669790893, 535756912504764218, 43471544417828923777, 3816784803681841133512, 360546156617986177328681, 36462349359125513109697520, 3930704977357944446111295571
Offset: 1
Two of the 5 tanglegrams for a(3) are illustrated (A,B are the roots of the trees and o marks the leaves that are shared between the two trees)
A A
/ \ / \
/ / \ / / \
o o o o o o
\ | / \ / /
\ | / \ /
B B
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\\ See links in A339645 for combinatorial species functions.
seriesReducedTrees(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n )); x*Ser(v)}
NumUnlabeledObjsSeq(sCartPower(seriesReducedTrees(15), 2))
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