A326565 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having the same sum.

1, 0, 1, 1, 4, 13, 91, 1318, 73581, 51913025

Offset

0,5

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-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

Links

Table of n, a(n) for n=0..9.

Example

The a(2) = 1 through a(5) = 13 antichains:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}      {{1,2,3,4,5}}
                      {{1,4},{2,3}}    {{1,2,5},{1,3,4}}
                      {{2,4},{1,2,3}}  {{1,3,5},{2,3,4}}
                      {{3,4},{1,2,4}}  {{1,4,5},{2,3,5}}
                                       {{1,4,5},{1,2,3,4}}
                                       {{2,3,5},{1,2,3,4}}
                                       {{2,4,5},{1,2,3,5}}
                                       {{3,4,5},{1,2,4,5}}
                                       {{1,5},{2,4},{1,2,3}}
                                       {{2,5},{3,4},{1,2,4}}
                                       {{3,5},{1,2,5},{1,3,4}}
                                       {{4,5},{1,3,5},{2,3,4}}
                                       {{1,4,5},{2,3,5},{1,2,3,4}}

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], {2, n}], SubsetQ[#1, #2]||Total[#1]!=Total[#2]&], Union@@#==Range[n]&];
Table[Length[cleq[n]], {n, 0, 5}]

Crossrefs

Cf. A000372, A006126, A035470, A307249, A321455, A321717, A321718, A326360, A326518, A326534, A326566.

Sequence in context: A131590 A281182 A200850 * A041433 A222764 A058014

Adjacent sequences: A326562 A326563 A326564 * A326566 A326567 A326568

Keyword

nonn,more

Author

Gus Wiseman, Jul 13 2019

Extensions

a(9) from Andrew Howroyd, Aug 14 2019

Status

approved