cp's OEIS Frontend

A384395 Number of integer partitions of n with more than one proper way to choose disjoint strict partitions of each part.

%I A384395 #5 May 30 2025 23:12:13
%S A384395 0,0,0,0,0,1,2,1,4,5,8,8,12,17,22,29,31,40,50,65,77,101,112,135,162,
%T A384395 201
%N A384395 Number of integer partitions of n with more than one proper way to choose disjoint strict partitions of each part.
%C A384395 By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.
%e A384395 For the partition (8,5,2) we have four choices:
%e A384395   ((8),(4,1),(2))
%e A384395   ((7,1),(5),(2))
%e A384395   ((5,3),(4,1),(2))
%e A384395   ((4,3,1),(5),(2))
%e A384395 Hence (8,5,2) is counted under a(15).
%e A384395 The a(5) = 1 through a(12) = 12 partitions:
%e A384395   (5)  (6)    (7)  (8)    (9)    (10)     (11)     (12)
%e A384395        (3,3)       (4,4)  (5,4)  (5,5)    (6,5)    (6,6)
%e A384395                    (5,3)  (6,3)  (6,4)    (7,4)    (7,5)
%e A384395                    (7,1)  (7,2)  (7,3)    (8,3)    (8,4)
%e A384395                           (8,1)  (8,2)    (9,2)    (9,3)
%e A384395                                  (9,1)    (10,1)   (10,2)
%e A384395                                  (4,3,3)  (5,3,3)  (11,1)
%e A384395                                  (4,4,2)  (5,5,1)  (5,5,2)
%e A384395                                                    (6,3,3)
%e A384395                                                    (6,4,2)
%e A384395                                                    (6,5,1)
%e A384395                                                    (9,2,1)
%t A384395 pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
%t A384395 Table[Length[Select[IntegerPartitions[n],Length[pofprop[#]]>1&]],{n,0,15}]
%Y A384395 For just one choice we have A179009, ranked by A383707.
%Y A384395 Twice-partitions of this type are counted by A279790.
%Y A384395 For at least one choice we have A383708, odd case A383533.
%Y A384395 For no choices we have A383710, odd case A383711.
%Y A384395 For at least one proper choice we have A384317, ranked by A384321.
%Y A384395 The strict version for at least one proper choice is A384318, ranked by A384322.
%Y A384395 The strict version for just one proper choice is A384319, ranked by A384390.
%Y A384395 For just one proper choice we have A384323, ranks A384347 = positions of 2 in A383706.
%Y A384395 For no proper choices we have A384348, ranked by A384349.
%Y A384395 These partitions are ranked by A384393.
%Y A384395 A000041 counts integer partitions, strict A000009.
%Y A384395 A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
%Y A384395 A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
%Y A384395 A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
%Y A384395 A357982 counts choices of strict partitions of each prime index, non-strict A299200.
%Y A384395 Cf. A098859, A279375, A317142, A381454, A382525, A382912, A382913, A384005.
%K A384395 nonn,more
%O A384395 0,7
%A A384395 _Gus Wiseman_, May 30 2025