A340022
Number of graphs with vertices labeled with positive integers summing to n.
Original entry on oeis.org
1, 1, 3, 7, 22, 71, 319, 1939, 19790, 377259, 14603435, 1144417513, 176665721300, 52525450429119, 29719386740326525, 31836493683553082697, 64474640381705842520802, 246962703426353769596309789, 1791765285568042699367722904797, 24670014908867411635732865067513309
Offset: 0
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permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
seq[n_] := 1 + Sum[s = 0; Do[s += permcount[p]*2^edges[p]*x^k/Product[1 - x^p[[j]] + O[x]^(n-k+1), {j, 1, Length[p]}],{p, IntegerPartitions[k]}]; s/k!, {k, 1, n}] // CoefficientList[#, x]&;
seq[19] (* Jean-François Alcover, Jan 06 2021, after Andrew Howroyd *)
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permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
seq(n) = {Vec(1+sum(k=1, n, my(s=0); forpart(p=k, s+=permcount(p) * 2^edges(p) * x^k/prod(j=1, #p, 1 - x^p[j] + O(x^(n-k+1)))); s/k!))}
A340026
Number of connected graphs with n integer labeled vertices covering an initial interval of positive integers.
Original entry on oeis.org
1, 1, 2, 12, 151, 3845, 192215, 18642053, 3534415032, 1322914720382, 983866402492022, 1458669558830420947, 4317992152324160500565, 25541957673530923214876165, 302031658361424323818453728818, 7141206379474081326199747144178588, 337646560987347470614138636684815527025
Offset: 0
a(2) = 2 because there is 1 connected graph on 2 vertices which can either have both vertices labeled 1 or one vertex labeled 1 and the other 2.
a(3) = 12 because there are 2 connected graphs on 3 vertices. The complete graph K_3 can be labeled in 4 ways (111, 112, 122, 123) and the path graph P_3 can be labeled in 8 ways (111, 112, 121, 122, 212, 123, 132, 213).
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\\ See A340023 for G(n,k).
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
seq(n)={my(v=concat([1], InvEulerT(vector(n, n, G(n, y))))); sum(k=0, n, subst(v, y, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))}
A340027
Number of inequivalent vertex colorings of connected graphs on n unlabeled vertices.
Original entry on oeis.org
1, 1, 2, 7, 50, 520, 10665, 400220, 29204589, 4143245857, 1146827743079, 619412332805088, 653237982066620540, 1346571060160843394520, 5432476352054378478159877, 42947950068987980977264834190, 666212968663987333085874313873428, 20301440661023158546856805172595805762
Offset: 0
a(3) = 7 because there are 2 connected graphs on 3 vertices. The complete graph K_3 can be coloring in 3 ways (111, 112, 123) and the path graph P_3 can be colored in 4 ways (111, 112, 121, 123).
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\\ See links in A339645 for combinatorial species functions.
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
graphsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 2^edges(p) * sMonomial(p)); s/n!}
graphsSeries(n)={sum(k=0, n, graphsCycleIndex(k)*x^k) + O(x*x^n)}
InequivalentColoringsSeq(1+sLog(graphsSeries(15)))
Showing 1-3 of 3 results.
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