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A384348 Number of integer partitions of n with no proper way to choose disjoint strict partitions of each part.

%I A384348 #5 May 30 2025 23:12:25
%S A384348 1,1,2,2,4,6,7,11,17,25,30,44,61,82,113,141,193,249,327,422,548,682,
%T A384348 881,1106,1400,1751
%N A384348 Number of integer partitions of n with no proper way to choose disjoint strict partitions of each part.
%C A384348 By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.
%e A384348 For the partition y = (5,4,2,1) we have the following proper ways to choose strict partitions of each part:
%e A384348   ((5),(3,1),(2),(1))
%e A384348   ((4,1),(4,2),(1))
%e A384348   ((4,1),(3,1),(2),(1))
%e A384348   ((3,2),(4),(2),(1))
%e A384348   ((3,2),(3,1),(2),(1))
%e A384348 But none of this is disjoint, so y is counted under a(12).
%e A384348 The a(1) = 1 through a(8) = 17 partitions:
%e A384348   (1)  (2)   (21)   (22)    (32)     (222)     (322)      (332)
%e A384348        (11)  (111)  (31)    (41)     (321)     (331)      (422)
%e A384348                     (211)   (221)    (411)     (421)      (431)
%e A384348                     (1111)  (311)    (2211)    (511)      (521)
%e A384348                             (2111)   (3111)    (2221)     (611)
%e A384348                             (11111)  (21111)   (3211)     (2222)
%e A384348                                      (111111)  (4111)     (3221)
%e A384348                                                (22111)    (3311)
%e A384348                                                (31111)    (4211)
%e A384348                                                (211111)   (5111)
%e A384348                                                (1111111)  (22211)
%e A384348                                                           (32111)
%e A384348                                                           (41111)
%e A384348                                                           (221111)
%e A384348                                                           (311111)
%e A384348                                                           (2111111)
%e A384348                                                           (11111111)
%t A384348 pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
%t A384348 Table[Length[Select[IntegerPartitions[n],Length[pofprop[#]]==0&]],{n,0,15}]
%Y A384348 The strict case is A179009, ranked by A383707.
%Y A384348 This is the proper version of A383710, odd case A383711.
%Y A384348 This is the proper complement of A383708, odd case A383533.
%Y A384348 The complement is counted by A384317, ranks A384321.
%Y A384348 The strict version for at least one proper choice is A384318, ranked by A384322.
%Y A384348 For just one proper choice we have A384319, ranked by A384390.
%Y A384348 For two choices we have A384323, ranks A384347 = positions of 2 in A383706.
%Y A384348 These partitions are ranked by A384349.
%Y A384348 For more than one proper choice we have A384395, ranked by A384393.
%Y A384348 A000041 counts integer partitions, strict A000009.
%Y A384348 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384348 A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
%Y A384348 A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
%Y A384348 Cf. A098859, A279790, A299200, A317142, A357982, A381454, A382525, A382912, A382913, A384005.
%K A384348 nonn
%O A384348 0,3
%A A384348 _Gus Wiseman_, May 30 2025