cp's OEIS Frontend

A326978 Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a weak antichain.

%I A326978 #4 Aug 13 2019 13:20:22
%S A326978 1,1,2,3,5,7,11,15,21,28,38,52,68,91,116,149,191,249,311,399,498,622,
%T A326978 773,971,1193,1478,1811,2222,2709,3311,4021,4882,5894,7110,8554,10273,
%U A326978 12312,14734,17578,20941,24905,29570,35056,41475,48983,57752,68025,79988
%N A326978 Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a weak antichain.
%C A326978 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}}.
%C A326978 A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other.
%e A326978 The a(0) = 1 through a(7) = 15 partitions:
%e A326978   ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A326978            (11)  (21)   (22)    (32)     (33)      (43)
%e A326978                  (111)  (31)    (41)     (42)      (52)
%e A326978                         (211)   (221)    (51)      (61)
%e A326978                         (1111)  (311)    (222)     (322)
%e A326978                                 (2111)   (321)     (331)
%e A326978                                 (11111)  (411)     (421)
%e A326978                                          (2211)    (511)
%e A326978                                          (3111)    (2221)
%e A326978                                          (21111)   (3211)
%e A326978                                          (111111)  (4111)
%e A326978                                                    (22111)
%e A326978                                                    (31111)
%e A326978                                                    (211111)
%e A326978                                                    (1111111)
%t A326978 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326978 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A326978 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326978 submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
%t A326978 Table[Length[Select[IntegerPartitions[n],stableQ[dual[primeMS/@#],submultQ]&]],{n,0,30}]
%Y A326978 Set-systems whose dual is a weak antichain are A326968.
%Y A326978 Factorizations whose dual is a weak antichain are A326975.
%Y A326978 The version where the dual is a strict antichain is A326977.
%Y A326978 Cf. A000041, A319564, A319728, A326966, A326969, A326970, A326971, A326973.
%K A326978 nonn
%O A326978 0,3
%A A326978 _Gus Wiseman_, Aug 13 2019