A327357 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of sets covering n vertices with non-spanning edge-connectivity k.

1, 0, 1, 1, 1, 4, 1, 3, 1, 30, 13, 33, 32, 6, 546, 421, 1302, 1915, 1510, 693, 316, 135, 45, 10, 1

Offset

0,6

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 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.

Links

Table of n, a(n) for n=0..24.

Example

Triangle begins:
    1
    0    1
    1    1
    4    1    3    1
   30   13   33   32    6
  546  421 1302 1915 1510  693  316  135   45   10    1
Row n = 3 counts the following antichains:
  {{1},{2,3}}    {{1,2,3}}  {{1,2},{1,3}}  {{1,2},{1,3},{2,3}}
  {{2},{1,3}}               {{1,2},{2,3}}
  {{3},{1,2}}               {{1,3},{2,3}}
  {{1},{2},{3}}

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[[1]]], Union@@s[[c[[1]]]]]]]]];
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]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1, 0, Length[sys]-Max@@Length/@Select[Union[Subsets[sys]], Length[csm[#]]!=1&]];
Table[Length[Select[stableSets[Subsets[Range[n], {1, n}], SubsetQ], Union@@#==Range[n]&&eConn[#]==k&]], {n, 0, 5}, {k, 0, 2^n}]//.{foe___, 0}:>{foe}

Crossrefs

Row sums are A307249.

Column k = 0 is A120338.

The non-covering version is A327353.

The version for spanning edge-connectivity is A327352.

The specialization to simple graphs is A327149, with unlabeled version A327201.

Cf. A014466, A293606, A326704, A326787, A327071, A327148, A327236, A327351.

Sequence in context: A039930 A123251 A368922 * A334810 A348566 A021246

Adjacent sequences: A327354 A327355 A327356 * A327358 A327359 A327360

Keyword

nonn,tabf,more

Author

Gus Wiseman, Sep 11 2019

Status

approved