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A322788
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Irregular triangle read by rows where if d|n then T(n,d) is the number of uniform multiset partitions of a multiset with d copies of each integer from 1 to n/d.
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4
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1, 2, 2, 2, 2, 5, 4, 3, 2, 2, 27, 11, 6, 4, 2, 2, 142, 29, 8, 4, 282, 12, 3, 1073, 101, 8, 4, 2, 2, 32034, 1581, 234, 75, 20, 6, 2, 2, 136853, 2660, 10, 4, 1527528, 1985, 91, 4, 4661087, 64596, 648, 20, 5, 2, 2, 227932993, 1280333, 41945, 231, 28, 6
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OFFSET
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1,2
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COMMENTS
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A multiset partition is uniform if all parts have the same size.
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1
2 2
2 2
5 4 3
2 2
27 11 6 4
2 2
142 29 8 4
282 12 3
1073 101 8 4
The multiset partitions counted under row 6:
{123456} {112233} {111222} {111111}
{123}{456} {112}{233} {111}{222} {111}{111}
{124}{356} {113}{223} {112}{122} {11}{11}{11}
{125}{346} {122}{133} {11}{12}{22} {1}{1}{1}{1}{1}{1}
{126}{345} {123}{123} {12}{12}{12}
{134}{256} {11}{22}{33} {1}{1}{1}{2}{2}{2}
{135}{246} {11}{23}{23}
{136}{245} {12}{12}{33}
{145}{236} {12}{13}{23}
{146}{235} {13}{13}{22}
{156}{234} {1}{1}{2}{2}{3}{3}
{12}{34}{56}
{12}{35}{46}
{12}{36}{45}
{13}{24}{56}
{13}{25}{46}
{13}{26}{45}
{14}{23}{56}
{14}{25}{36}
{14}{26}{35}
{15}{23}{46}
{15}{24}{36}
{15}{26}{34}
{16}{23}{45}
{16}{24}{35}
{16}{25}{34}
{1}{2}{3}{4}{5}{6}
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MATHEMATICA
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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]]]];
Table[Length[Select[mps[Join@@Table[Range[n/d], {d}]], SameQ@@Length/@#&]], {n, 10}, {d, Divisors[n]}]
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CROSSREFS
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Cf. A001055, A005176, A056239, A072774, A100778, A295193, A306017, A319190, A319612, A322784, A322785, A322786, A322789, A322792.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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