A330475 Number of balanced reduced multisystems whose atoms constitute a strongly normal multiset of size n.

1, 1, 2, 9, 85, 1143, 25270

Offset

0,3

Comments

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

A finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

Links

Table of n, a(n) for n=0..6.

Example

The a(0) = 1 through a(3) = 9 multisystems:
  {}  {1}  {1,1}  {1,1,1}
           {1,2}  {1,1,2}
                  {1,2,3}
                  {{1},{1,1}}
                  {{1},{1,2}}
                  {{1},{2,3}}
                  {{2},{1,1}}
                  {{2},{1,3}}
                  {{3},{1,2}}

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
totm[m_]:=Prepend[Join@@Table[totm[p], {p, Select[mps[m], 1<Length[#]<Length[m]&]}], m];
Table[Sum[Length[totm[m]], {m, strnorm[n]}], {n, 0, 5}]

Crossrefs

The (weakly) normal version is A330655.

The maximum-depth case is A330675.

The case where the atoms are {1..n} is A005121.

The case where the atoms are all 1's is A318813.

The tree version is A330471.

Multiset partitions of strongly normal multisets are A035310.

Cf. A000311, A000669, A001678, A316652, A318812, A330467, A330474, A330628, A330663, A330679.

Sequence in context: A120959 A125797 A068595 * A356615 A037172 A106163

Adjacent sequences: A330472 A330473 A330474 * A330476 A330477 A330478

Keyword

nonn,more

Author

Gus Wiseman, Dec 27 2019

Status

approved