A340025
Number of connected graphs with vertices labeled with positive integers summing to n.
Original entry on oeis.org
1, 1, 2, 4, 12, 41, 210, 1478, 17128, 352926, 14181309, 1129005180, 175491164826, 52346463432414, 29666505555854777, 31806668884174645842, 64442744342933382243031, 246898165053174167804654086, 1791518193851453375966274280997, 24668222649527263942329934357240780
Offset: 0
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\\ See A340022 for permcount, edges.
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
seq(n) = {concat([1], InvEulerT(Vec(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)))}
A340024
Number of inequivalent vertex colorings of graphs on n unlabeled vertices.
Original entry on oeis.org
1, 1, 4, 14, 89, 788, 13712, 459380, 31395800, 4304547500, 1170501781632, 626269787446920, 657129205489027200, 1350883625562244545584, 5441806297331472273603040, 42987375826579901036722653600, 666538741644051928632441002162384, 20306710978262167791045247702178986496
Offset: 0
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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(graphsSeries(15))
Showing 1-3 of 3 results.
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