A327357 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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:

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