Displaying 1-10 of 22 results found.
Number of 2-edge-connected labeled graphs on n nodes.
+10
38
0, 0, 0, 1, 10, 253, 11968, 1047613, 169181040, 51017714393, 29180467201536, 32121680070545657, 68867078000231169536, 290155435185687263172693, 2417761175748567327193407488, 40013922635723692336670167608181, 1318910073755307133701940625759574016
COMMENTS
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
MATHEMATICA
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 *)
PROG
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
CROSSREFS
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.
AUTHOR
Yifei Chen (yifei(AT)mit.edu), Jul 17 2004
Number of labeled simple connected graphs with n vertices and at least one bridge, or graphs with spanning edge-connectivity 1.
+10
27
0, 0, 1, 3, 28, 475, 14736, 818643, 82367552, 15278576679, 5316021393280, 3519977478407687, 4487518206535452672, 11116767463976825779115, 53887635281876408097483776, 513758302006787897939587736715, 9668884580476067306398361085853696
COMMENTS
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.
MATHEMATICA
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}]
CROSSREFS
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.
BII-numbers of set-systems with spanning edge-connectivity 1.
+10
22
1, 2, 4, 5, 6, 7, 8, 16, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 56, 57, 58, 59, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 88, 89, 90, 91, 96, 97, 98, 99
COMMENTS
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges. The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.
EXAMPLE
The sequence of all set-systems with spanning edge-connectivity 1 together with their BII-numbers begins:
1: {{1}}
2: {{2}}
4: {{1,2}}
5: {{1},{1,2}}
6: {{2},{1,2}}
7: {{1},{2},{1,2}}
8: {{3}}
16: {{1,3}}
17: {{1},{1,3}}
20: {{1,2},{1,3}}
21: {{1},{1,2},{1,3}}
22: {{2},{1,2},{1,3}}
23: {{1},{2},{1,2},{1,3}}
24: {{3},{1,3}}
25: {{1},{3},{1,3}}
28: {{1,2},{3},{1,3}}
29: {{1},{1,2},{3},{1,3}}
30: {{2},{1,2},{3},{1,3}}
31: {{1},{2},{1,2},{3},{1,3}}
32: {{2,3}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
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&];
Select[Range[0, 100], spanEdgeConn[Union@@bpe/@bpe[#], bpe/@bpe[#]]==1&]
CROSSREFS
Graphs with spanning edge-connectivity >= 2 are counted by A095983. BII-numbers for vertex-connectivity 1 are A327098. BII-numbers for non-spanning edge-connectivity 1 are A327099. BII-numbers for spanning edge-connectivity 2 are A327108. BII-numbers for spanning edge-connectivity >= 2 are A327109. Set-systems with spanning edge-connectivity 2 are counted by A327130. Graphs with spanning edge-connectivity 1 are counted by A327145. Graphs with spanning edge-connectivity 2 are counted by A327146.
Number of n-node connected labeled graphs without endpoints.
+10
21
1, 1, 0, 1, 10, 253, 12058, 1052443, 169488200, 51045018089, 29184193354806, 32122530765469967, 68867427921051098084, 290155706369032525823085, 2417761578629525173499004146, 40013923790443379076988789688611, 1318910080173114018084245406769861936
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 404.
FORMULA
a(n) = Sum_{i=0..n} (-1)^i*binomial(n, i)*c(n-i)*(n-i)^i, for n>2, a(0)=1, a(1)=1, a(2)=0, where c(n) is number of n-node connected labeled graphs (cf. A001187). E.g.f.: 1 + x^2/2 + log(Sum_{n >= 0} 2^binomial(n, 2)*(x*exp(-x))^n/n!).
MAPLE
c:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
add(k*binomial(n, k)*2^((n-k)*(n-k-1)/2)*c(k), k=1..n-1)/n)
end:
a:= n-> max(0, add((-1)^i*binomial(n, i)*c(n-i)*(n-i)^i, i=0..n)):
MATHEMATICA
Flatten[{1, 1, 0, Table[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}], {n, 3, 15}]}] (* Vaclav Kotesovec, May 14 2015 *) c[0] = 1; c[n_] := c[n] = 2^(n*(n-1)/2) - Sum[k*Binomial[n, k]*2^((n-k)*(n - k - 1)/2)*c[k], {k, 1, n-1}]/n; a[0] = a[1] = 1; a[2] = 0; a[n_] := Sum[(-1)^i*Binomial[n, i]*c[n-i]*(n-i)^i, {i, 0, n}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 27 2017, using Alois P. Heinz's code for c(n) *)
PROG
(PARI) seq(n)={Vec(serlaplace(1 + x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!))))} \\ Andrew Howroyd, Sep 09 2018
CROSSREFS
Cf. A059167 (n-node labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A004110 (n-node unlabeled graphs without endpoints).
EXTENSIONS
More terms from John Renze (jrenze(AT)yahoo.com), Feb 01 2001
Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.
+10
17
1, 1, 1, 1, 1, 3, 3, 1, 4, 18, 27, 14, 1, 56, 250, 402, 240, 65, 10, 1, 1031, 5475, 11277, 9620, 4282, 921, 146, 15, 1
COMMENTS
The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any isolated vertices) to obtain a disconnected or empty graph.
FORMULA
T(n,k) = Sum_{m = 0..n} binomial(n,m) A327149(m,k). In words, column k is the binomial transform of column k of A327149.
EXAMPLE
Triangle begins:
1
1
1 1
1 3 3 1
4 18 27 14 1
56 250 402 240 65 10 1
MATHEMATICA
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]]]]]]]]];
edgeConnSys[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}]], edgeConnSys[#]==k&]], {n, 0, 4}, {k, 0, Binomial[n, 2]}]//.{foe___, 0}:>{foe}
CROSSREFS
The corresponding triangle for vertex-connectivity is A327125. The corresponding triangle for spanning edge-connectivity is A327069. Cf. A001187, A263296, A322338, A322395, A326787, A327079, A327097, A327099, A327102, A327126, A327144, A327196, A327200, A327201.
Number of labeled n-vertex graphs without vertices of degree <=1.
+10
14
1, 0, 0, 1, 10, 253, 12068, 1052793, 169505868, 51046350021, 29184353055900, 32122563615242615, 68867440268165982320, 290155715157676330952559, 2417761590648159731258579164, 40013923822242935823157820555477, 1318910080336893719646370269435043184
FORMULA
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
EXAMPLE
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)
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Graphs without isolated nodes are A006129. Graphs without endpoints are A059167. Labeled graphs with endpoints are A245797.
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).
+10
14
0, 0, 1, 0, 12, 180, 4200, 157920, 9673664, 1011129840, 190600639200, 67674822473280, 46325637863907072, 61746583700640860736, 161051184122415878112640, 824849999242893693424992000, 8317799170120961768715123118080
COMMENTS
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.
FORMULA
Inverse binomial transform of A327231.
MATHEMATICA
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}]
CROSSREFS
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.
BII-numbers of set-systems with non-spanning edge-connectivity 2.
+10
13
5, 6, 17, 20, 24, 34, 36, 40, 48, 53, 54, 55, 60, 61, 62, 63, 65, 66, 68, 71, 72, 80, 86, 87, 89, 92, 93, 94, 95, 96, 101, 103, 106, 108, 109, 110, 111, 113, 114, 115, 121, 122, 123, 257, 260, 272, 308, 309, 310, 311, 316, 317, 318, 319, 320, 326, 327, 342
COMMENTS
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges. The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any isolated vertices) to result in a disconnected or empty set-system.
EXAMPLE
The sequence of all set-systems with non-spanning edge-connectivity 2 together with their BII-numbers begins:
5: {{1},{1,2}}
6: {{2},{1,2}}
17: {{1},{1,3}}
20: {{1,2},{1,3}}
24: {{3},{1,3}}
34: {{2},{2,3}}
36: {{1,2},{2,3}}
40: {{3},{2,3}}
48: {{1,3},{2,3}}
53: {{1},{1,2},{1,3},{2,3}}
54: {{2},{1,2},{1,3},{2,3}}
55: {{1},{2},{1,2},{1,3},{2,3}}
60: {{1,2},{3},{1,3},{2,3}}
61: {{1},{1,2},{3},{1,3},{2,3}}
62: {{2},{1,2},{3},{1,3},{2,3}}
63: {{1},{2},{1,2},{3},{1,3},{2,3}}
65: {{1},{1,2,3}}
66: {{2},{1,2,3}}
68: {{1,2},{1,2,3}}
71: {{1},{2},{1,2},{1,2,3}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
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]]]]]]]]];
edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1, 0, Length[y]-Max@@Length/@Select[Union[Subsets[y]], Length[csm[bpe/@#]]!=1&]];
Select[Range[0, 100], edgeConn[bpe[#]]==2&]
CROSSREFS
BII-numbers for vertex-connectivity 2 are A327082. BII-numbers for non-spanning edge-connectivity 1 are A327099. BII-numbers for non-spanning edge-connectivity > 1 are A327102. BII-numbers for spanning edge-connectivity 2 are A327108. Cf. A007146, A048793, A052446, A059166, A070939, A095983, A263296, A322335, A322338, A322395, A326031, A327041, A327069, A327111.
BII-numbers of set-systems with non-spanning edge-connectivity >= 2.
+10
11
5, 6, 17, 20, 21, 24, 34, 36, 38, 40, 48, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 72, 80, 81, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 98, 100, 101, 102, 103, 104, 106, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121
COMMENTS
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges. A set-system has non-spanning 2-edge-connectivity >= 2 if it is connected and any single edge can be removed (along with any non-covered vertices) without making the set-system disconnected or empty. Alternatively, these are connected set-systems whose bridges (edges whose removal disconnects the set-system or leaves isolated vertices) are all endpoints (edges intersecting only one other edge).
EXAMPLE
The sequence of all set-systems with non-spanning edge-connectivity >= 2 together with their BII-numbers begins:
5: {{1},{1,2}}
6: {{2},{1,2}}
17: {{1},{1,3}}
20: {{1,2},{1,3}}
21: {{1},{1,2},{1,3}}
24: {{3},{1,3}}
34: {{2},{2,3}}
36: {{1,2},{2,3}}
38: {{2},{1,2},{2,3}}
40: {{3},{2,3}}
48: {{1,3},{2,3}}
52: {{1,2},{1,3},{2,3}}
53: {{1},{1,2},{1,3},{2,3}}
54: {{2},{1,2},{1,3},{2,3}}
55: {{1},{2},{1,2},{1,3},{2,3}}
56: {{3},{1,3},{2,3}}
60: {{1,2},{3},{1,3},{2,3}}
61: {{1},{1,2},{3},{1,3},{2,3}}
62: {{2},{1,2},{3},{1,3},{2,3}}
63: {{1},{2},{1,2},{3},{1,3},{2,3}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
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]]]]]]]]];
edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1, 0, Length[y]-Max@@Length/@Select[Union[Subsets[y]], Length[csm[bpe/@#]]!=1&]];
Select[Range[0, 100], edgeConn[bpe[#]]>=2&]
CROSSREFS
Graphs with spanning edge-connectivity >= 2 are counted by A095983. Graphs with non-spanning edge-connectivity >= 2 are counted by A322395. Also positions of terms >=2 in A326787. BII-numbers for non-spanning edge-connectivity 2 are A327097. BII-numbers for non-spanning edge-connectivity 1 are A327099. BII-numbers for spanning edge-connectivity >= 2 are A327109. Cf. A000120, A048793, A059166, A070939, A263296, A326031, A326749, A327076, A327101, A327102, A327108, A327148.
Number of connected set-systems covering n vertices with at least one edge whose removal (along with any non-covered vertices) disconnects the set-system (non-spanning edge-connectivity 1).
+10
9
COMMENTS
A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.
FORMULA
Inverse binomial transform of A327196.
EXAMPLE
The a(3) = 35 set-systems:
{123} {1}{12}{23} {1}{2}{12}{13} {1}{2}{3}{12}{13}
{1}{13}{23} {1}{2}{12}{23} {1}{2}{3}{12}{23}
{1}{2}{123} {1}{2}{13}{23} {1}{2}{3}{13}{23}
{1}{3}{123} {1}{2}{3}{123} {1}{2}{3}{12}{123}
{2}{12}{13} {1}{3}{12}{13} {1}{2}{3}{13}{123}
{2}{13}{23} {1}{3}{12}{23} {1}{2}{3}{23}{123}
{2}{3}{123} {1}{3}{13}{23}
{3}{12}{13} {2}{3}{12}{13}
{3}{12}{23} {2}{3}{12}{23}
{1}{23}{123} {2}{3}{13}{23}
{2}{13}{123} {1}{2}{13}{123}
{3}{12}{123} {1}{2}{23}{123}
{1}{3}{12}{123}
{1}{3}{23}{123}
{2}{3}{12}{123}
{2}{3}{13}{123}
MATHEMATICA
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], {1, n}]], Union@@#==Range[n]&&eConn[#]==1&]], {n, 0, 3}]
CROSSREFS
The restriction to simple graphs is A327079, with non-covering version A327231. The version for spanning edge-connectivity is A327145, with BII-numbers A327111. The BII-numbers of these set-systems are A327099. The non-covering version is A327196.
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