This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A327351 Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity exactly k. 11
 1, 1, 0, 1, 1, 0, 4, 3, 2, 0, 30, 40, 27, 17, 0, 546, 1365, 1842, 1690, 1451, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices. The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0. If empty edges are allowed, we have T(0,0) = 2. LINKS EXAMPLE Triangle begins:     1     1    0     1    1    0     4    3    2    0    30   40   27   17    0   546 1365 1842 1690 1451    0 MATHEMATICA csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[]], Union@@s[[c[]]]]]]]]; 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]]]]; vertConnSys[vts_, eds_]:=Min@@Length/@Select[Subsets[vts], Function[del, Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds, Alternatives@@del, {2}], {}]]!={Complement[vts, del]}]] Table[Length[Select[stableSets[Subsets[Range[n], {1, n}], SubsetQ], Union@@#==Range[n]&&vertConnSys[Range[n], #]==k&]], {n, 0, 4}, {k, 0, n}] CROSSREFS Row sums are A307249, or A006126 if empty edges are allowed. Column k = 0 is A120338, if we assume A120338(0) = A120338(1) = 1. Column k = 1 is A327356. Column k = n - 1 is A327020. The unlabeled version is A327359. The version for vertex-connectivity >= k is A327350. The version for spanning edge-connectivity is A327352. The version for non-spanning edge-connectivity is A327353, with covering case A327357. Cf. A003465, A006126, A014466, A048143, A293993, A323818, A326704, A327125, A327334, A327336. Sequence in context: A258692 A067018 A200233 * A239475 A100802 A022960 Adjacent sequences:  A327348 A327349 A327350 * A327352 A327353 A327354 KEYWORD nonn,tabl,more AUTHOR Gus Wiseman, Sep 09 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 21 16:04 EST 2019. Contains 329371 sequences. (Running on oeis4.)