cp's OEIS Frontend

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.

%I A326977 #6 Aug 13 2019 13:20:15
%S A326977 1,1,2,3,5,7,10,14,20,27,36,49,64,85,109,141,181,234,294,375,470,589,
%T A326977 733,917,1131,1401,1720,2113,2581,3153,3833,4655,5631,6801,8192,9849,
%U A326977 11816,14148,16899,20153,23990,28503,33815,40038,47330,55858,65841,77475
%N 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.
%C A326977 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 A326977 The a(0) = 1 through a(7) = 14 partitions:
%e A326977   ()  (1)  (2)   (3)    (4)     (5)      (33)      (7)
%e A326977            (11)  (21)   (22)    (32)     (42)      (43)
%e A326977                  (111)  (31)    (41)     (51)      (52)
%e A326977                         (211)   (221)    (222)     (322)
%e A326977                         (1111)  (311)    (321)     (331)
%e A326977                                 (2111)   (411)     (421)
%e A326977                                 (11111)  (2211)    (511)
%e A326977                                          (3111)    (2221)
%e A326977                                          (21111)   (3211)
%e A326977                                          (111111)  (4111)
%e A326977                                                    (22111)
%e A326977                                                    (31111)
%e A326977                                                    (211111)
%e A326977                                                    (1111111)
%t A326977 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
%t A326977 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A326977 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326977 submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
%t A326977 Table[Length[Select[IntegerPartitions[n],UnsameQ@@dual[primeMS/@#]&&stableQ[dual[primeMS/@#],submultQ]&]],{n,0,30}]
%Y A326977 T_0 integer partitions are A319564.
%Y A326977 Set-systems whose dual is a (strict) antichain are A326965.
%Y A326977 The version where the dual is a weak antichain is A326978.
%Y A326977 T_1 factorizations (whose dual is a strict antichain) are A327012.
%Y A326977 Cf. A000041, A319728, A326961, A326974, A326976, A326979.
%K A326977 nonn
%O A326977 0,3
%A A326977 _Gus Wiseman_, Aug 13 2019