A324171 Number of non-crossing multiset partitions of normal multisets of size n.

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

Offset: 0

Author

Gus Wiseman, Feb 17 2019

Keywords

• nonn
• more

Comments

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

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.

Examples

			The A255906(5) - a(5) = 22 crossing multiset partitions:
  {{13}{124}}  {{1}{13}{24}}
  {{13}{224}}  {{1}{24}{35}}
  {{13}{234}}  {{2}{13}{24}}
  {{13}{244}}  {{2}{14}{35}}
  {{13}{245}}  {{3}{13}{24}}
  {{14}{235}}  {{3}{14}{25}}
  {{24}{113}}  {{4}{13}{24}}
  {{24}{123}}  {{4}{13}{25}}
  {{24}{133}}  {{5}{13}{24}}
  {{24}{134}}
  {{24}{135}}
  {{25}{134}}
  {{35}{124}}
		

Crossrefs

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

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

Programs

Mathematica

nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Sum[Length[Select[mps[m],nonXQ]],{m,allnorm[n]}],{n,0,8}]