OFFSET
0,2
COMMENTS
An antichain is non-crossing if no pair of distinct parts is of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
FORMULA
Binomial transform of A324167.
G.f.: A(x) = B(x/(1-2*x))/x where B(x)/x is the g.f. of A359984. - Andrew Howroyd, Jan 20 2023
EXAMPLE
The a(0) = 1 through a(3) = 19 non-crossing antichains:
{} {} {} {}
{{1}} {{1}} {{1}}
{{2}} {{2}}
{{12}} {{3}}
{{1}{2}} {{12}}
{{13}}
{{23}}
{{123}}
{{1}{2}}
{{1}{3}}
{{2}{3}}
{{1}{23}}
{{2}{13}}
{{3}{12}}
{{12}{13}}
{{12}{23}}
{{13}{23}}
{{1}{2}{3}}
{{12}{13}{23}}
MATHEMATICA
nn=6;
nonXQ[stn_]:=!MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<y<t||z<x<t<y];
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]]]];
Table[Length[stableSets[Subsets[Range[n], {1, n}], SubsetQ[##]||!nonXQ[{#1, #2}]&]], {n, 0, nn}]
PROG
(PARI) seq(n)={my(f=O(1)); for(n=2, n, f = 1 + (4*x + x^2)*f^2 - 3*x^2*(1 + x)*f^3); Vec(subst(x*(1 + x^2*f^2 - 3*x^3*f^3), x, x/(1-2*x))/x) } \\ Andrew Howroyd, Jan 20 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 17 2019
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Jan 20 2023
STATUS
approved