A326277 Number of crossing normal multiset partitions of weight n.

0, 0, 0, 0, 1, 22, 314, 3711, 39947

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.

Links

Table of n, a(n) for n=0..8.

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

Crossing simple graphs are A326210.

Normal multiset partitions are A255906.

Non-crossing normal multiset partitions are A324171.

MM-numbers of crossing multiset partitions are A324170.

Cf. A000108, A016098, A054726, A058681.

Cf. A326211, A326212, A326255, A326256, A326258.

Sequence in context: A025992 A028034 A028231 * A025988 A267132 A023949

Adjacent sequences: A326274 A326275 A326276 * A326278 A326279 A326280

Keyword

nonn,more

Author

Gus Wiseman, Jun 22 2019

Status

approved