A330679 Number of balanced reduced multisystems whose atoms constitute an integer partition of n.

1, 1, 2, 4, 12, 40, 180, 936, 5820, 41288, 331748, 2968688, 29307780, 316273976, 3704154568, 46788812168, 634037127612, 9174782661984, 141197140912208, 2302765704401360, 39671953757409256, 719926077632193848, 13726066030661998220, 274313334040504957368

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Formula

a(n > 1) = 2 * A318813(n).

Example

The a(0) = 1 through a(4) = 12 multisystems:
  {}  {1}  {2}    {3}          {4}
           {1,1}  {1,2}        {1,3}
                  {1,1,1}      {2,2}
                  {{1},{1,1}}  {1,1,2}
                               {1,1,1,1}
                               {{1},{1,2}}
                               {{2},{1,1}}
                               {{1},{1,1,1}}
                               {{1,1},{1,1}}
                               {{1},{1},{1,1}}
                               {{{1}},{{1},{1,1}}}
                               {{{1,1}},{{1},{1}}}

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

Crossrefs

The case where the atoms are all 1's is A318813 = a(n)/2.

The version where the atoms constitute a strongly normal multiset is A330475.

The version where the atoms cover an initial interval is A330655.

The maximum-depth version is A330726.

Cf. A000041, A000111, A000669, A001970, A002846, A005121, A141268, A196545, A213427, A318812, A320160, A330474.

Sequence in context: A062962 A033472 A134983 * A342225 A218144 A222919

Adjacent sequences: A330676 A330677 A330678 * A330680 A330681 A330682

Keyword

nonn

Author

Gus Wiseman, Dec 31 2019

Extensions

a(12) onwards from Andrew Howroyd, Jan 20 2024

Status

approved