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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
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.
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.
Sequence in context: A114597 A199479 A050165 * A198876 A033878 A144061
KEYWORD
nonn,more,tabl
AUTHOR
Gus Wiseman, Jan 03 2020
STATUS
approved