cp's OEIS Frontend

A324167 Number of non-crossing antichain covers of {1,...,n}.

%I A324167 #9 Jan 20 2023 22:50:13
%S A324167 1,1,2,9,67,633,6763,77766,938957,11739033,150649945,1973059212,
%T A324167 26265513030,354344889798,4833929879517,66568517557803,
%U A324167 924166526830701,12920482325488761,181750521972603049,2570566932237176232,36532394627404815308,521439507533582646156
%N A324167 Number of non-crossing antichain covers of {1,...,n}.
%C A324167 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.
%H A324167 Andrew Howroyd, <a href="/A324167/b324167.txt">Table of n, a(n) for n = 0..500</a>
%F A324167 Inverse binomial transform of A324168.
%F A324167 Binomial transform of A359984. - _Andrew Howroyd_, Jan 20 2023
%e A324167 The a(3) = 9 antichains:
%e A324167   {{1,2,3}}
%e A324167   {{1},{2,3}}
%e A324167   {{2},{1,3}}
%e A324167   {{3},{1,2}}
%e A324167   {{1,2},{1,3}}
%e A324167   {{1,2},{2,3}}
%e A324167   {{1,3},{2,3}}
%e A324167   {{1},{2},{3}}
%e A324167   {{1,2},{1,3},{2,3}}
%t A324167 nn=6;
%t A324167 croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A324167 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]]]];
%t A324167 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ[##]||croXQ[{#1,#2}]&],Union@@#==Range[n]&]],{n,0,nn}]
%o A324167 (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-x))/x) } \\ _Andrew Howroyd_, Jan 20 2023
%Y A324167 Cf. A000108, A000124, A000372 (antichains), A001006, A006126 (antichain covers), A014466, A048143, A054726 (non-crossing graphs), A099947, A261005, A283877, A306438.
%Y A324167 Cf. A324166, A324168, A324169, A324170, A324171, A324173, A359984 (no singletons).
%K A324167 nonn
%O A324167 0,3
%A A324167 _Gus Wiseman_, Feb 17 2019
%E A324167 Terms a(9) and beyond from _Andrew Howroyd_, Jan 20 2023