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A326977
Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a (strict) antichain, also called T_1 integer partitions.
14
1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 49, 64, 85, 109, 141, 181, 234, 294, 375, 470, 589, 733, 917, 1131, 1401, 1720, 2113, 2581, 3153, 3833, 4655, 5631, 6801, 8192, 9849, 11816, 14148, 16899, 20153, 23990, 28503, 33815, 40038, 47330, 55858, 65841, 77475
OFFSET
0,3
COMMENTS
The dual of a multiset partition has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set of multisets, none of which is a submultiset of any other.
EXAMPLE
The a(0) = 1 through a(7) = 14 partitions:
() (1) (2) (3) (4) (5) (33) (7)
(11) (21) (22) (32) (42) (43)
(111) (31) (41) (51) (52)
(211) (221) (222) (322)
(1111) (311) (321) (331)
(2111) (411) (421)
(11111) (2211) (511)
(3111) (2221)
(21111) (3211)
(111111) (4111)
(22111)
(31111)
(211111)
(1111111)
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}];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
submultQ[cap_, fat_]:=And@@Function[i, Count[fat, i]>=Count[cap, i]]/@Union[List@@cap];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@dual[primeMS/@#]&&stableQ[dual[primeMS/@#], submultQ]&]], {n, 0, 30}]
CROSSREFS
T_0 integer partitions are A319564.
Set-systems whose dual is a (strict) antichain are A326965.
The version where the dual is a weak antichain is A326978.
T_1 factorizations (whose dual is a strict antichain) are A327012.
Sequence in context: A145786 A094023 A123630 * A035967 A097797 A219601
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 13 2019
STATUS
approved