|
|
A326359
|
|
Number of maximal antichains of nonempty subsets of {1..n}.
|
|
13
|
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
A set system (set of sets) is an antichain if no element is a subset of any other.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
The a(0) = 1 through a(4) = 28 antichains:
{} {1} {12} {123} {1234}
{1}{2} {1}{23} {1}{234}
{2}{13} {2}{134}
{3}{12} {3}{124}
{1}{2}{3} {4}{123}
{12}{13}{23} {1}{2}{34}
{1}{3}{24}
{1}{4}{23}
{2}{3}{14}
{2}{4}{13}
{3}{4}{12}
{1}{2}{3}{4}
{12}{134}{234}
{13}{124}{234}
{14}{123}{234}
{23}{124}{134}
{24}{123}{134}
{34}{123}{124}
{1}{23}{24}{34}
{2}{13}{14}{34}
{3}{12}{14}{24}
{4}{12}{13}{23}
{12}{13}{14}{234}
{12}{23}{24}{134}
{13}{23}{34}{124}
{14}{24}{34}{123}
{123}{124}{134}{234}
{12}{13}{14}{23}{24}{34}
|
|
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]]]];
fasmax[y_]:=Complement[y, Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n], {1, n}], SubsetQ]]], {n, 0, 5}]
|
|
CROSSREFS
|
Antichains of nonempty sets are A014466.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of sets are A326358.
Cf. A000372, A003182, A006126, A006602, A014466, A058891, A261005, A305000, A305844, A307249, A326360, A326362, A326363.
|
|
KEYWORD
|
nonn,more
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|