|
%I #5 Sep 03 2019 09:57:53
%S 1,1,1,1,1,1,1,1,2,2,3,3,1,4,5,10,8,5,1,1
%N Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs with n vertices whose edge-set has non-spanning edge-connectivity k.
%C The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a disconnected or empty graph, ignoring isolated vertices.
%H Gus Wiseman, <a href="/A327236/a327236.png">Unlabeled graphs with 5 vertices, organized by non-spanning edge-connectivity (isolated vertices not shown).</a>
%e Triangle begins:
%e 1
%e 1
%e 1 1
%e 1 1 1 1
%e 2 2 3 3 1
%e 4 5 10 8 5 1 1
%t 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]]]]]]]]];
%t edgeConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
%t Table[Length[Union[normclut/@Select[Subsets[Subsets[Range[n],{2}]],edgeConnSys[#]==k&]]],{n,0,5},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}
%Y Row sums are A000088.
%Y Column k = 0 is A327235.
%Y The labeled version is A327148.
%Y The covering version is A327201.
%Y Spanning edge-connectivity is A263296.
%Y Vertex-connectivity is A259862.
%Y Cf. A322338, A322396, A326787, A327069, A327077, A327097, A327099, A327102, A327200, A327231.
%K nonn,tabf,more
%O 0,9
%A _Gus Wiseman_, Sep 03 2019
|
