A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.

1, 1, 1, 1, 3, 2, 1, 5, 9, 5, 1, 9, 28, 36, 16, 1, 13, 69, 160, 164, 61, 1, 20, 160, 580, 1022, 855, 272, 1, 28, 337, 1837, 4996, 7072, 4988, 1385

Offset

2,5

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

Table of n, a(n) for n=2..37.

Formula

T(n,3) = A000041(n) - 2.

T(n,4) = A001970(n) - 3 * A000041(n) + 3.

Example

Triangle begins:
    1
    1    1
    1    3    2
    1    5    9    5
    1    9   28   36   16
    1   13   69  160  164   61
    1   20  160  580 1022  855  272
    1   28  337 1837 4996 7072 4988 1385
Row n = 5 counts the following multisystems (strings of 1's are replaced by their lengths):
  5  {1,4}      {{1},{1,3}}      {{{1}},{{1},{1,2}}}
     {2,3}      {{1},{2,2}}      {{{1,1}},{{1},{2}}}
     {1,1,3}    {{2},{1,2}}      {{{1}},{{2},{1,1}}}
     {1,2,2}    {{3},{1,1}}      {{{1,2}},{{1},{1}}}
     {1,1,1,2}  {{1},{1,1,2}}    {{{2}},{{1},{1,1}}}
                {{1,1},{1,2}}
                {{2},{1,1,1}}
                {{1},{1},{1,2}}
                {{1},{2},{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[Length[Select[totm[ConstantArray[1, n]], Depth[#]==k&]], {n, 2, 6}, {k, 2, n}]

Crossrefs

Row sums are A318813.

Column k = 3 is A007042.

Column k = 4 is A001970(n) - 3*A000041(n) + 3.

Column k = n is A000111.

Row n is row prime(n) of A330727.

Cf. A000669, A001055, A002846, A005121, A196545, A213427, A318812, A320160, A330474, A330475, A330655, A330667, A330679.

Sequence in context: A114597 A199479 A050165 * A198876 A033878 A144061

Adjacent sequences: A330781 A330782 A330783 * A330785 A330786 A330787

Keyword

nonn,more,tabl

Author

Gus Wiseman, Jan 03 2020

Status

approved