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A322788
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.
4
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
OFFSET
1,2
COMMENTS
A multiset partition is uniform if all parts have the same size.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..482 (rows 1..100)
FORMULA
T(n,d) = A322794(A002110(n/d)^d).
EXAMPLE
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}
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]]]];
Table[Length[Select[mps[Join@@Table[Range[n/d], {d}]], SameQ@@Length/@#&]], {n, 10}, {d, Divisors[n]}]
CROSSREFS
KEYWORD
nonn,look,tabf
AUTHOR
Gus Wiseman, Dec 26 2018
EXTENSIONS
More terms from Alois P. Heinz, Jan 30 2019
Terms a(38) and beyond from Andrew Howroyd, Feb 03 2022
STATUS
approved