A319728 Number of strict T_0 integer partitions of n.

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).

Links

Table of n, a(n) for n=0..55.

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}]

Crossrefs

Cf. A000009, A000041, A001970, A007716, A059201, A305148, A316983, A319558, A319564, A319616.

Sequence in context: A018053 A146930 A333524 * A241152 A367843 A164529

Adjacent sequences: A319725 A319726 A319727 * A319729 A319730 A319731

Keyword

nonn

Author

Gus Wiseman, Sep 26 2018

Status

approved