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A384323 Number of integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.

%I A384323 #8 May 30 2025 23:14:24
%S A384323 0,0,0,1,1,0,3,3,2,0,6,6,6,6,4,10,10,14,16,15,16,17,20,25,27,28,37,43,
%T A384323 31,42,44
%N A384323 Number of integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.
%e A384323 For y = (4,3,3) we have two ways: ((4),(3),(2,1)) and ((4),(2,1),(3)), so y is counted under a(10).
%e A384323 The a(0) = 0 through a(15) = 10 partitions:
%e A384323   .  .  .  3  4  .  33  43  44  .  433  533  543  544  554  5433
%e A384323                     42  52  62     442  542  552  553  644  5442
%e A384323                     51  61         532  551  633  652  662  5532
%e A384323                                    541  632  732  661  833  5541
%e A384323                                    631  731  741  733       6432
%e A384323                                    721  821  831  832       6531
%e A384323                                                             7431
%e A384323                                                             7521
%e A384323                                                             8421
%e A384323                                                             9321
%t A384323 pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
%t A384323 Table[Length[Select[IntegerPartitions[n],Length[pof[#]]==2&]],{n,0,15}]
%Y A384323 For just one choice we have A179009, ranked by A383707.
%Y A384323 Twice-partitions of this type are counted by A279790.
%Y A384323 For at least one choice we have A383708, odd case A383533.
%Y A384323 For no choices we have A383710, odd case A383711.
%Y A384323 For more than one choice we have A384317, ranked by A384321.
%Y A384323 The strict version for at least one choice is A384318, ranked by A384322.
%Y A384323 The strict version is A384319, ranked by A384390.
%Y A384323 These partitions are ranked by A384347 = positions of 2 in A383706.
%Y A384323 A000041 counts integer partitions, strict A000009.
%Y A384323 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384323 A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
%Y A384323 A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
%Y A384323 A357982 counts choices of strict partitions of each prime index.
%Y A384323 Cf. A098859, A299200, A317142, A381454, A382525, A382912, A382913, A384005.
%K A384323 nonn,more
%O A384323 0,7
%A A384323 _Gus Wiseman_, May 30 2025