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A319728
Number of strict T_0 integer partitions of n.
5
1, 1, 1, 2, 2, 3, 3, 4, 6, 8, 9, 10, 14, 16, 19, 25, 31, 34, 41, 49, 59, 72, 81, 94, 113, 133, 152, 179, 209, 239, 273, 315, 366, 422, 478, 548, 627, 711, 812, 926, 1051, 1185, 1340, 1514, 1718, 1945, 2179, 2444, 2757, 3095, 3465, 3892, 4362, 4865, 5427, 6068
OFFSET
0,4
COMMENTS
The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. For an integer partition the T_0 condition means the dual of the multiset partition obtained by factoring each part into prime numbers is strict (no repeated blocks).
EXAMPLE
The a(11) = 10 integer partitions are (11), (7,4), (8,3), (9,2), (5,4,2), (6,3,2), (6,4,1), (7,3,1), (8,2,1), (5,3,2,1). Missing from this list are (6,5) and (10,1).
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}]
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&UnsameQ@@dual[primeMS/@#]&]], {n, 60}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 26 2018
STATUS
approved