cp's OEIS Frontend

A327353 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of subsets of {1..n} with non-spanning edge-connectivity k.

%I A327353 #7 Sep 11 2019 20:21:53
%S A327353 1,1,1,2,3,8,7,3,1,53,27,45,36,6,747,511,1497,2085,1540,693,316,135,
%T A327353 45,10,1
%N A327353 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of subsets of {1..n} with non-spanning edge-connectivity k.
%C A327353 An antichain is a set of sets, none of which is a subset of any other.
%C A327353 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.
%e A327353 Triangle begins:
%e A327353     1
%e A327353     1    1
%e A327353     2    3
%e A327353     8    7    3    1
%e A327353    53   27   45   36    6
%e A327353   747  511 1497 2085 1540  693  316  135   45   10    1
%e A327353 Row n = 3 counts the following antichains:
%e A327353   {}             {{1}}      {{1,2},{1,3}}  {{1,2},{1,3},{2,3}}
%e A327353   {{1},{2}}      {{2}}      {{1,2},{2,3}}
%e A327353   {{1},{3}}      {{3}}      {{1,3},{2,3}}
%e A327353   {{2},{3}}      {{1,2}}
%e A327353   {{1},{2,3}}    {{1,3}}
%e A327353   {{2},{1,3}}    {{2,3}}
%e A327353   {{3},{1,2}}    {{1,2,3}}
%e A327353   {{1},{2},{3}}
%t A327353 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 A327353 stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
%t A327353 eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
%t A327353 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],eConn[#]==k&]],{n,0,4},{k,0,2^n}]//.{foe___,0}:>{foe}
%Y A327353 Row sums are A014466.
%Y A327353 Column k = 0 is A327354.
%Y A327353 The covering case is A327357.
%Y A327353 The version for spanning edge-connectivity is A327352.
%Y A327353 The specialization to simple graphs is A327148, with covering case A327149, unlabeled version A327236, and unlabeled covering case A327201.
%Y A327353 Cf. A052446, A307249, A326704, A326787, A327071, A327351, A327355.
%K A327353 nonn,tabf,more
%O A327353 0,4
%A A327353 _Gus Wiseman_, Sep 10 2019