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A326277 Number of crossing normal multiset partitions of weight n.

%I A326277 #5 Jun 22 2019 23:14:35
%S A326277 0,0,0,0,1,22,314,3711,39947
%N A326277 Number of crossing normal multiset partitions of weight n.
%C A326277 A multiset partition is normal if it covers an initial interval of positive integers.
%C A326277 A multiset partition is crossing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < y < t or z < x < t < y.
%e A326277 The a(5) = 22 crossing normal multiset partitions:
%e A326277   {{1,3},{1,2,4}}  {{1},{1,3},{2,4}}
%e A326277   {{1,3},{2,2,4}}  {{1},{2,4},{3,5}}
%e A326277   {{1,3},{2,3,4}}  {{2},{1,3},{2,4}}
%e A326277   {{1,3},{2,4,4}}  {{2},{1,4},{3,5}}
%e A326277   {{1,3},{2,4,5}}  {{3},{1,3},{2,4}}
%e A326277   {{1,4},{2,3,5}}  {{3},{1,4},{2,5}}
%e A326277   {{2,4},{1,1,3}}  {{4},{1,3},{2,4}}
%e A326277   {{2,4},{1,2,3}}  {{4},{1,3},{2,5}}
%e A326277   {{2,4},{1,3,3}}  {{5},{1,3},{2,4}}
%e A326277   {{2,4},{1,3,4}}
%e A326277   {{2,4},{1,3,5}}
%e A326277   {{2,5},{1,3,4}}
%e A326277   {{3,5},{1,2,4}}
%t A326277 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A326277 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A326277 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A326277 croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A326277 Table[Length[Select[Join@@mps/@allnorm[n],croXQ]],{n,0,6}]
%Y A326277 Crossing simple graphs are A326210.
%Y A326277 Normal multiset partitions are A255906.
%Y A326277 Non-crossing normal multiset partitions are A324171.
%Y A326277 MM-numbers of crossing multiset partitions are A324170.
%Y A326277 Cf. A000108, A016098, A054726, A058681.
%Y A326277 Cf. A326211, A326212, A326255, A326256, A326258.
%K A326277 nonn,more
%O A326277 0,6
%A A326277 _Gus Wiseman_, Jun 22 2019