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A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified July 27 15:29 EDT 2021. Contains 346307 sequences. (Running on oeis4.)