A326570 Number of covering antichains of subsets of {1..n} with different edge-sizes.

2, 1, 1, 4, 17, 186, 3292, 139161, 14224121

Offset

0,1

Comments

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sizes are the numbers of vertices in each edge, so for example the edge-sizes of {{1,3},{2,5},{3,4,5}} are {2,2,3}.

Links

Table of n, a(n) for n=0..8.

Example

The a(0) = 2 through a(4) = 17 antichains:
  {}    {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}
  {{}}                  {{1},{2,3}}  {{1},{2,3,4}}
                        {{2},{1,3}}  {{2},{1,3,4}}
                        {{3},{1,2}}  {{3},{1,2,4}}
                                     {{4},{1,2,3}}
                                     {{1,2},{1,3,4}}
                                     {{1,2},{2,3,4}}
                                     {{1,3},{1,2,4}}
                                     {{1,3},{2,3,4}}
                                     {{1,4},{1,2,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{1,2,4}}
                                     {{2,3},{1,3,4}}
                                     {{2,4},{1,2,3}}
                                     {{2,4},{1,3,4}}
                                     {{3,4},{1,2,3}}
                                     {{3,4},{1,2,4}}

Mathematica

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]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n]], SubsetQ[#1, #2]||Length[#1]==Length[#2]&], Union@@#==Range[n]&];
Table[Length[cleq[n]], {n, 0, 6}]

Crossrefs

Antichain covers are A006126.

Set partitions with different block sizes are A007837.

The case without singletons is A326569.

(Antichain) covers with equal edge-sizes are A306021.

Cf. A000372, A003182, A307249, A326026, A326514, A326566, A326572.

Sequence in context: A083677 A075803 A127966 * A184880 A257859 A231622

Adjacent sequences: A326567 A326568 A326569 * A326571 A326572 A326573

Keyword

nonn,more

Author

Gus Wiseman, Jul 18 2019

Extensions

a(8) from Andrew Howroyd, Aug 13 2019

Status

approved