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%I #6 Aug 13 2019 13:20:15
%S 1,1,2,3,5,7,10,14,20,27,36,49,64,85,109,141,181,234,294,375,470,589,
%T 733,917,1131,1401,1720,2113,2581,3153,3833,4655,5631,6801,8192,9849,
%U 11816,14148,16899,20153,23990,28503,33815,40038,47330,55858,65841,77475
%N 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.
%C 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.
%e The a(0) = 1 through a(7) = 14 partitions:
%e () (1) (2) (3) (4) (5) (33) (7)
%e (11) (21) (22) (32) (42) (43)
%e (111) (31) (41) (51) (52)
%e (211) (221) (222) (322)
%e (1111) (311) (321) (331)
%e (2111) (411) (421)
%e (11111) (2211) (511)
%e (3111) (2221)
%e (21111) (3211)
%e (111111) (4111)
%e (22111)
%e (31111)
%e (211111)
%e (1111111)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@dual[primeMS/@#]&&stableQ[dual[primeMS/@#],submultQ]&]],{n,0,30}]
%Y T_0 integer partitions are A319564.
%Y Set-systems whose dual is a (strict) antichain are A326965.
%Y The version where the dual is a weak antichain is A326978.
%Y T_1 factorizations (whose dual is a strict antichain) are A327012.
%Y Cf. A000041, A319728, A326961, A326974, A326976, A326979.
%K nonn
%O 0,3
%A _Gus Wiseman_, Aug 13 2019