cp's OEIS Frontend

A384317 Number of integer partitions of n with more than one possible way to choose disjoint strict partitions of each part.

%I A384317 #6 May 28 2025 09:17:21
%S A384317 0,0,0,1,1,1,4,4,5,5,12,12,16,19,22,35,38,48,58,68,79,110,121,149,175,
%T A384317 207,242,281,352,397,473
%N A384317 Number of integer partitions of n with more than one possible way to choose disjoint strict partitions of each part.
%C A384317 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%F A384317 a(n) = A383708(n) - A179009(n).
%e A384317 There are two possibilities for (4,3), namely ((4),(3)) and ((4),(2,1)), so (4,3) is counted under a(7).
%e A384317 The a(3) = 1 through a(11) = 12 partitions:
%e A384317   (3)  (4)  (5)  (6)    (7)    (8)    (9)    (10)     (11)
%e A384317                  (3,3)  (4,3)  (4,4)  (5,4)  (5,5)    (6,5)
%e A384317                  (4,2)  (5,2)  (5,3)  (6,3)  (6,4)    (7,4)
%e A384317                  (5,1)  (6,1)  (6,2)  (7,2)  (7,3)    (8,3)
%e A384317                                (7,1)  (8,1)  (8,2)    (9,2)
%e A384317                                              (9,1)    (10,1)
%e A384317                                              (4,3,3)  (5,3,3)
%e A384317                                              (4,4,2)  (5,4,2)
%e A384317                                              (5,3,2)  (5,5,1)
%e A384317                                              (5,4,1)  (6,3,2)
%e A384317                                              (6,3,1)  (7,3,1)
%e A384317                                              (7,2,1)  (8,2,1)
%t A384317 pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
%t A384317 Table[Length[Select[IntegerPartitions[n],Length[pof[#]]>1&]],{n,0,30}]
%Y A384317 The case of a unique choice is A179009, ranks A383707.
%Y A384317 The case of at least one choice is A383708, ranks A382913.
%Y A384317 The case of no choices is A383710, ranks A382912.
%Y A384317 The strict case is A384318, ranks A384322.
%Y A384317 These partitions are ranked by A384321, positions of terms > 1 in A383706.
%Y A384317 The case of a unique proper choice is A384323, ranks A384347, strict A384319.
%Y A384317 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384317 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A384317 A357982 counts choices of strict partitions of prime indices, non-strict A299200.
%Y A384317 Cf. A098859, A381454, A382525, A383533, A383711, A384320.
%K A384317 nonn,more
%O A384317 0,7
%A A384317 _Gus Wiseman_, May 28 2025