A324171 - OEIS

%I #7 Feb 19 2019 00:08:17

%S 1,1,4,16,75,378,2042,11489,66697

%N Number of non-crossing multiset partitions of normal multisets of size n.

%C A multiset is normal if its union is an initial interval of positive integers.

%C A multiset partition is crossing if it has a 2-element submultiset of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

%e The A255906(5) - a(5) = 22 crossing multiset partitions:

%e   {{13}{124}}  {{1}{13}{24}}

%e   {{13}{224}}  {{1}{24}{35}}

%e   {{13}{234}}  {{2}{13}{24}}

%e   {{13}{244}}  {{2}{14}{35}}

%e   {{13}{245}}  {{3}{13}{24}}

%e   {{14}{235}}  {{3}{14}{25}}

%e   {{24}{113}}  {{4}{13}{24}}

%e   {{24}{123}}  {{4}{13}{25}}

%e   {{24}{133}}  {{5}{13}{24}}

%e   {{24}{134}}

%e   {{24}{135}}

%e   {{25}{134}}

%e   {{35}{124}}

%t nonXQ[stn_]:=!MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];

%t Table[Sum[Length[Select[mps[m],nonXQ]],{m,allnorm[n]}],{n,0,8}]

%Y Cf. A000108 (non-crossing set partitions), A000124, A001006, A001055, A001263, A007297, A054726 (non-crossing graphs), A099947, A194560, A255906 (multiset partitions of normal multisets), A306438.

%Y Cf. A324166, A324167, A324168, A324169, A324170, A324173.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Feb 17 2019