OFFSET
0,6
COMMENTS
A multiset partition is normal if it covers an initial interval of positive integers.
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.
EXAMPLE
The a(5) = 22 crossing normal multiset partitions:
{{1,3},{1,2,4}} {{1},{1,3},{2,4}}
{{1,3},{2,2,4}} {{1},{2,4},{3,5}}
{{1,3},{2,3,4}} {{2},{1,3},{2,4}}
{{1,3},{2,4,4}} {{2},{1,4},{3,5}}
{{1,3},{2,4,5}} {{3},{1,3},{2,4}}
{{1,4},{2,3,5}} {{3},{1,4},{2,5}}
{{2,4},{1,1,3}} {{4},{1,3},{2,4}}
{{2,4},{1,2,3}} {{4},{1,3},{2,5}}
{{2,4},{1,3,3}} {{5},{1,3},{2,4}}
{{2,4},{1,3,4}}
{{2,4},{1,3,5}}
{{2,5},{1,3,4}}
{{3,5},{1,2,4}}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=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]];
croXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<y<t||z<x<t<y];
Table[Length[Select[Join@@mps/@allnorm[n], croXQ]], {n, 0, 6}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jun 22 2019
STATUS
approved