login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A326570 Number of covering antichains of subsets of {1..n} with different edge-sizes. 4
2, 1, 1, 4, 17, 186, 3292, 139161, 14224121 (list; graph; refs; listen; history; text; internal format)
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

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.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 25 11:32 EST 2020. Contains 338623 sequences. (Running on oeis4.)