cp's OEIS Frontend

A327149 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of simple labeled graphs covering n vertices with non-spanning edge-connectivity k.

%I A327149 #10 Feb 06 2023 09:59:02
%S A327149 1,0,1,0,0,3,1,3,12,15,10,1,40,180,297,180,60,10,1
%N A327149 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of simple labeled graphs covering n vertices with non-spanning edge-connectivity k.
%C A327149 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.
%F A327149 A327148(n,k) = Sum_{m = 0..n} binomial(n,m) T(m,k). In words, column k is the inverse binomial transform of column k of A327148.
%e A327149 Triangle begins:
%e A327149    1
%e A327149    {}
%e A327149    0   1
%e A327149    0   0   3   1
%e A327149    3  12  15  10   1
%e A327149   40 180 297 180  60  10   1
%t A327149 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]]]]]]]]];
%t A327149 eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
%t A327149 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&eConn[#]==k&]],{n,0,4},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}
%Y A327149 Row sums are A006129.
%Y A327149 Column k = 0 is A327070.
%Y A327149 Column k = 1 is A327079.
%Y A327149 The corresponding triangle for vertex-connectivity is A327126.
%Y A327149 The corresponding triangle for spanning edge-connectivity is A327069.
%Y A327149 The non-covering version is A327148.
%Y A327149 The unlabeled version is A327201.
%Y A327149 Cf. A001187, A263296, A322338, A322395, A326787, A327097, A327099, A327102, A327125, A327129, A327144.
%K A327149 nonn,tabf,more
%O A327149 0,6
%A A327149 _Gus Wiseman_, Aug 27 2019