Search: a059166 -id:a059166
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A095983
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Number of 2-edge-connected labeled graphs on n nodes.
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+10
38
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0, 0, 0, 1, 10, 253, 11968, 1047613, 169181040, 51017714393, 29180467201536, 32121680070545657, 68867078000231169536, 290155435185687263172693, 2417761175748567327193407488, 40013922635723692336670167608181, 1318910073755307133701940625759574016
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OFFSET
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0,5
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COMMENTS
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Essentially the same sequence arises as the number of connected bridgeless labeled graphs (graphs that are k-edge connected for k >= 2, starting elements of this sequence are 1, 1, 0, 1, 10, 253, 11968, ...).
The spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a graph that is disconnected or covers fewer vertices. This sequence counts graphs with spanning edge-connectivity >= 2, which, for n > 1, are connected graphs with no bridges. - Gus Wiseman, Sep 20 2019
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LINKS
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FORMULA
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MATHEMATICA
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seq[n_] := Module[{v, p, q, c}, v[_] = 0; p = x*D[#, x]& @ Log[ Sum[ 2^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^(n+1)]; q = x*E^p; p -= q; For[k = 3, k <= n, k++, c = Coefficient[p, x, k]; v[k] = c*(k-1)!; p -= c*q^k]; Join[{0}, Array[v, n]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
spanEdgeConn[vts_, eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds], Union@@#!=vts||Length[csm[#]]!=1&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], spanEdgeConn[Range[n], #]>=2&]], {n, 0, 5}] (* Gus Wiseman, Sep 20 2019 *)
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PROG
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seq(n)={my(v=vector(n));
my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))));
my(q=x*exp(p)); p-=q;
for(k=3, n, my(c=polcoeff(p, k)); v[k]=c*(k-1)!; p-=c*q^k);
(PARI) seq(n)={my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))); Vec(serlaplace(log(x/serreverse(x*exp(p)))/x-1), -(n+1))} \\ Andrew Howroyd, Dec 28 2020
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CROSSREFS
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Row sums of A327069 if the first two columns are removed.
BII-numbers of set-systems with spanning edge-connectivity >= 2 are A327109.
Graphs with spanning edge-connectivity 2 are A327146.
Graphs with non-spanning edge-connectivity >= 2 are A327200.
2-vertex-connected graphs are A013922.
Graphs without endpoints are A059167.
Graphs with spanning edge-connectivity 1 are A327071.
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KEYWORD
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nonn
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AUTHOR
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Yifei Chen (yifei(AT)mit.edu), Jul 17 2004
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EXTENSIONS
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STATUS
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approved
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A004110
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Number of n-node unlabeled graphs without endpoints (i.e., no nodes of degree 1).
(Formerly M1504)
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+10
33
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1, 1, 1, 2, 5, 16, 78, 588, 8047, 205914, 10014882, 912908876, 154636289460, 48597794716736, 28412296651708628, 31024938435794151088, 63533059372622888758054, 244916078509480823407040988, 1783406527599529094009748567708
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OFFSET
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COMMENTS
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a(n) is also the number of unlabeled mating graphs with n nodes. A mating graph has no two vertices with identical sets of neighbors. - Tanya Khovanova, Oct 23 2008
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REFERENCES
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F. Harary, Graph Theory, Wiley, 1969. See illustrations in Appendix 1.
F. Harary and E. Palmer, Graphical Enumeration, (1973), compare formula (8.7.11).
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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MATHEMATICA
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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]];
a[n_] := Sum[permcount[p] * 2^edges[p] * Coefficient[Product[1 - x^p[[i]], {i, 1, Length[p]}], x, n - k]/k!, {k, 1, n}, {p, IntegerPartitions[k]}]; a[0] = 1;
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PROG
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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)}
a(n) = {my(s=0); sum(k=1, n, forpart(p=k, s+=permcount(p) * 2^edges(p) * polcoef(prod(i=1, #p, 1-x^p[i]), n-k)/k!)); s} \\ Andrew Howroyd, Sep 09 2018
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CROSSREFS
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Cf. A059166 (n-node connected labeled graphs without endpoints), A059167 (n-node labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A006024 (number of labeled mating graphs with n nodes), A129584 (bi-point-determining graphs).
If isolated nodes are forbidden, see A261919.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A059167
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Number of n-node labeled graphs without endpoints.
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+10
30
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1, 1, 1, 2, 15, 314, 13757, 1142968, 178281041, 52610850316, 29702573255587, 32446427369694348, 69254848513798160815, 291053505824567573585744, 2421830049319361003822380177, 40050220743831370293688592267252, 1319550593412053164173947687592553185
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OFFSET
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0,4
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REFERENCES
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F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 31, problem 1.16(a).
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LINKS
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FORMULA
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a(n) = Sum_{i=0..n-1} binomial(n-1, i)*b(i+1)*a(n-i-1), n>0, a(0)=1, where b(n) is number of n-node connected labeled graphs without endpoints (Cf. A059166).
E.g.f.: exp(x^2/2)*(Sum_{n >= 0} 2^binomial(n, 2)*(x/exp(x))^n/n!). - Vladeta Jovovic, Mar 23 2004
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MAPLE
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F:= proc(N) local S;
S:= series(exp(1/2*x^2)*Sum(2^binomial(n, 2)*(x/exp(x))^n/n!, n = 0 .. N), x, N+1);
seq(coeff(S, x, i)*i!, i=0..N)
end proc:
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MATHEMATICA
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b[n_] := If[n < 3, Boole[n == 1], n!*Sum[(-1)^(n - j) * SeriesCoefficient[1 + Log[Sum[2^(k*(k - 1)/2)*x^k/k!, {k, 0, j}]], {x, 0, j}] * j^(n - j)/(n - j)!, {j, 0, n}]]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, i] b[i + 1] a[n - i - 1], {i, 0, n - 1}]; Table[a@ n, {n, 0, 15}] (* Michael De Vlieger, Sep 19 2016, after Vaclav Kotesovec at A059166 *)
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PROG
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(PARI) seq(n)={my(A=x/exp(x + O(x^n))); Vec(serlaplace(exp(x^2/2 + O(x*x^n)) * sum(k=0, n, 2^binomial(k, 2)*A^k/k!)))} \\ Andrew Howroyd, Sep 09 2018
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CROSSREFS
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Cf. A059166 (n-node connected labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A004110 (n-node unlabeled graphs without endpoints).
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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More terms from John Renze (jrenze(AT)yahoo.com), Feb 01 2001
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STATUS
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approved
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A327071
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Number of labeled simple connected graphs with n vertices and at least one bridge, or graphs with spanning edge-connectivity 1.
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+10
27
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0, 0, 1, 3, 28, 475, 14736, 818643, 82367552, 15278576679, 5316021393280, 3519977478407687, 4487518206535452672, 11116767463976825779115, 53887635281876408097483776, 513758302006787897939587736715, 9668884580476067306398361085853696
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OFFSET
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0,4
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COMMENTS
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A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Connected graphs with no bridges are counted by A095983 (2-edge-connected graphs).
The spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty graph.
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LINKS
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FORMULA
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MATHEMATICA
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csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
spanEdgeConn[vts_, eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds], Union@@#!=vts||Length[csm[#]]!=1&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], spanEdgeConn[Range[n], #]==1&]], {n, 0, 4}]
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CROSSREFS
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Connected graphs without bridges are A007146.
The enumeration of labeled connected graphs by number of bridges is A327072.
Connected graphs with exactly one bridge are A327073.
Graphs with non-spanning edge-connectivity 1 are A327079.
BII-numbers of set-systems with spanning edge-connectivity 1 are A327111.
Covering set-systems with spanning edge-connectivity 1 are A327145.
Graphs with spanning edge-connectivity 2 are A327146.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A004108
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Number of n-node unlabeled connected graphs without endpoints.
(Formerly M2910)
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+10
24
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1, 1, 0, 1, 3, 11, 61, 507, 7442, 197772, 9808209, 902884343, 153723152913, 48443147912137, 28363697921914475, 30996525982586676021, 63502034385272108655525, 244852545421597419740767106, 1783161611489937453151313949442, 24603891216883233547700609793901996
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OFFSET
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0,5
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COMMENTS
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Also number of n-node unlabeled connected mating graphs, cf. A006024 and A092430 (conjectured by Vladeta Jovovic, proved by G. Kilibarda). - Vladeta Jovovic, Oct 07 2004
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REFERENCES
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F. Harary and E. Palmer, Graphical Enumeration, (1973), formula (8.7.11).
Goran Kilibarda, "Enumeration of unlabeled mating graphs", Belgrade, 2004, to be published.
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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MATHEMATICA
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terms = 19;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
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]];
b[n_] := Sum[permcount[p]*2^edges[p]*Coefficient[Product[1 - x^p[[i]], {i, 1, Length[p]}], x, n - k]/k!, {k, 1, n}, {p, IntegerPartitions[k]}];
A004110 = Table[b[n], {n, 1, terms-1}];
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CROSSREFS
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Cf. A059166 (n-node connected labeled graphs without endpoints), A059167 (n-node labeled graphs without endpoints), A004110 (Euler Transform, n-node unlabeled graphs without endpoints).
Cf. A092430 (n-node labeled connected mating graphs).
Counts include those for distance-critical graphs, A349402.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A245797
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The number of labeled graphs of n vertices that have endpoints, where an endpoint is a vertex with degree 1.
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+10
22
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0, 1, 6, 49, 710, 19011, 954184, 90154415, 16108626420, 5481798833245, 3582369649269620, 4532127781040045649, 11177949079089720090800, 54050029251399545975868271, 514598463471970554205910304780, 9677402372862708729859372687791391
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OFFSET
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1,3
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LINKS
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FORMULA
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Binomial transform of A327227 (assuming a(0) = 0).
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MATHEMATICA
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m = 16;
egf = Exp[x^2/2]*Sum[2^Binomial[n, 2]*(x/Exp[x])^n/n!, {n, 0, m}];
A059167[n_] := SeriesCoefficient[egf, {x, 0, n}]*n!;
a[n_] := 2^(n(n-1)/2) - A059167[n];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Min@@Length/@Split[Sort[Join@@#]]==1&]], {n, 0, 5}] (* Gus Wiseman, Sep 11 2019 *)
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CROSSREFS
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The generalization to set-systems is A327228.
BII-numbers of set-systems with minimum degree 1 are A327105.
Cf. A001187, A006129, A059166, A059167, A100743, A136284, A327079, A327098, A327103, A327229, A327230.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A327227
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Number of labeled simple graphs covering n vertices with at least one endpoint/leaf.
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+10
22
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0, 0, 1, 3, 31, 515, 15381, 834491, 83016613, 15330074139, 5324658838645, 3522941267488973, 4489497643961740521, 11119309286377621015089, 53893949089393110881259181, 513788884660608277842596504415, 9669175277199248753133328740702449
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OFFSET
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COMMENTS
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Covering means there are no isolated vertices.
A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.
Also graphs with minimum vertex-degree 1.
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LINKS
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FORMULA
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EXAMPLE
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The a(4) = 31 edge-sets:
{12,34} {12,13,14} {12,13,14,23}
{13,24} {12,13,24} {12,13,14,24}
{14,23} {12,13,34} {12,13,14,34}
{12,14,23} {12,13,23,24}
{12,14,34} {12,13,23,34}
{12,23,24} {12,14,23,24}
{12,23,34} {12,14,24,34}
{12,24,34} {12,23,24,34}
{13,14,23} {13,14,23,34}
{13,14,24} {13,14,24,34}
{13,23,24} {13,23,24,34}
{13,23,34} {14,23,24,34}
{13,24,34}
{14,23,24}
{14,23,34}
{14,24,34}
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MATHEMATICA
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Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]==1&]], {n, 0, 5}]
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CROSSREFS
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The non-covering version is A245797.
The generalization to set-systems is A327229.
BII-numbers of set-systems with minimum degree 1 are A327105.
Cf. A001187, A006129, A059166, A059167, A100743, A136284, A327079, A327098, A327103, A327228, A327230.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A100743
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Number of labeled n-vertex graphs without vertices of degree <=1.
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+10
14
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1, 0, 0, 1, 10, 253, 12068, 1052793, 169505868, 51046350021, 29184353055900, 32122563615242615, 68867440268165982320, 290155715157676330952559, 2417761590648159731258579164, 40013923822242935823157820555477, 1318910080336893719646370269435043184
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OFFSET
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0,5
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LINKS
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FORMULA
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E.g.f.: exp(-x+x^2/2)*(Sum_{n>=0} 2^(n*(n-1)/2)*(x/exp(x))^n/n!). - Vladeta Jovovic, Jan 26 2006
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EXAMPLE
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The a(4) = 10 edge-sets:
{12,13,24,34}
{12,14,23,34}
{13,14,23,24}
{12,13,14,23,24}
{12,13,14,23,34}
{12,13,14,24,34}
{12,13,23,24,34}
{12,14,23,24,34}
{13,14,23,24,34}
{12,13,14,23,24,34}
(End)
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MATHEMATICA
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m = 13;
egf = Exp[-x + x^2/2]*Sum[2^(n (n-1)/2)*(x/Exp[x])^n/n!, {n, 0, m+1}];
s = egf + O[x]^(m+1);
a[n_] := n!*SeriesCoefficient[s, n];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]>1&]], {n, 0, 4}] (* Gus Wiseman, Aug 18 2019 *)
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PROG
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(PARI) seq(n)={Vec(serlaplace(exp(-x + x^2/2 + O(x*x^n))*sum(k=0, n, 2^(k*(k-1)/2)*(x/exp(x + O(x^n)))^k/k!)))} \\ Andrew Howroyd, Sep 04 2019
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CROSSREFS
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Graphs without isolated nodes are A006129.
Graphs without endpoints are A059167.
Labeled graphs with endpoints are A245797.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A327079
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Number of labeled simple connected graphs covering n vertices with at least one bridge that is not an endpoint/leaf (non-spanning edge-connectivity 1).
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+10
14
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0, 0, 1, 0, 12, 180, 4200, 157920, 9673664, 1011129840, 190600639200, 67674822473280, 46325637863907072, 61746583700640860736, 161051184122415878112640, 824849999242893693424992000, 8317799170120961768715123118080
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OFFSET
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0,5
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COMMENTS
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A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Graphs with no bridges are counted by A095983 (2-edge-connected graphs).
Also labeled simple connected graphs covering n vertices with non-spanning edge-connectivity 1, where the non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty graph.
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LINKS
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FORMULA
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Inverse binomial transform of A327231.
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MATHEMATICA
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csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1, 0, Length[sys]-Max@@Length/@Select[Union[Subsets[sys]], Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&eConn[#]==1&]], {n, 0, 4}]
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CROSSREFS
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The non-covering version is A327231.
Connected bridged graphs (spanning edge-connectivity 1) are A327071.
BII-numbers of graphs with non-spanning edge-connectivity 1 are A327099.
Covering set-systems with non-spanning edge-connectivity 1 are A327129.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A327369
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Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).
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+10
14
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1, 1, 0, 1, 0, 1, 2, 0, 6, 0, 15, 12, 30, 4, 3, 314, 320, 260, 80, 50, 0, 13757, 10890, 5445, 1860, 735, 66, 15, 1142968, 640836, 228564, 64680, 16800, 2772, 532, 0, 178281041, 68362504, 17288852, 3666600, 702030, 115416, 17892, 1016, 105
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,7
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LINKS
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FORMULA
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Column-wise binomial transform of A327377.
E.g.f.: exp(x + U(x,y) + B(x*(1-y) + R(x,y))), where R(x,y) is the e.g.f. of A055302, U(x,y) is the e.g.f. of A055314 and B(x) + x is the e.g.f. of A059167. - Andrew Howroyd, Oct 05 2019
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EXAMPLE
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Triangle begins:
1
1 0
1 0 1
2 0 6 0
15 12 30 4 3
314 320 260 80 50 0
13757 10890 5445 1860 735 66 15
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MATHEMATICA
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Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Count[Length/@Split[Sort[Join@@#]], 1]==k&]], {n, 0, 5}, {k, 0, n}]
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PROG
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(PARI)
my(R=sum(n=1, n, x^n*sum(k=1, n, stirling(n-1, n-k, 2)*y^k/k!)) + O(x*x^n));
my(U=sum(n=2, n, x^n*sum(k=1, n, stirling(n-2, n-k, 2)*y^k/k!)) + O(x*x^n));
my(B=x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!)));
my(A=exp(x + U + subst(B-x, x, x*(1-y) + R)));
Vecrev(n!*polcoef(A, n), n + 1);
}
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CROSSREFS
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Row sums without the first column are A245797.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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