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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 (list; graph; refs; listen; history; text; internal format)
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 A164529 A153906

Adjacent sequences:  A319725 A319726 A319727 * A319729 A319730 A319731

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 26 2018

STATUS

approved

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Last modified April 10 06:23 EDT 2020. Contains 333392 sequences. (Running on oeis4.)