cp's OEIS Frontend

A387328 Number of integer partitions of n whose parts have choosable sets of integer partitions.

%I A387328 #6 Sep 02 2025 21:57:31
%S A387328 1,1,1,2,3,4,5,7,10,13,17,22,28,36,46,58,73,91,114,141,174,214,262,
%T A387328 320,389,472,571,688,828,993,1189,1419,1690,2009,2383,2821,3334,3931,
%U A387328 4628,5439,6381,7474,8741,10207,11902,13858,16114,18710,21698,25130,29070
%N A387328 Number of integer partitions of n whose parts have choosable sets of integer partitions.
%C A387328 First differs from A052335 at A052335(20) = 173, a(20) = 174, corresponding to the partition (4,4,4,4,4).
%C A387328 a(n) is the number of integer partitions of n such that it is possible to choose a sequence of distinct integer partitions, one of each part.
%C A387328 Also the number of integer partitions y of n with no part k whose multiplicity in y exceeds A000041(k).
%e A387328 The a(1) = 1 through a(9) = 13 partitions:
%e A387328   (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
%e A387328             (21)  (22)  (32)   (33)   (43)   (44)    (54)
%e A387328                   (31)  (41)   (42)   (52)   (53)    (63)
%e A387328                         (221)  (51)   (61)   (62)    (72)
%e A387328                                (321)  (322)  (71)    (81)
%e A387328                                       (331)  (332)   (333)
%e A387328                                       (421)  (422)   (432)
%e A387328                                              (431)   (441)
%e A387328                                              (521)   (522)
%e A387328                                              (3221)  (531)
%e A387328                                                      (621)
%e A387328                                                      (3321)
%e A387328                                                      (4221)
%t A387328 Table[Length[Select[IntegerPartitions[n],Select[Tuples[IntegerPartitions/@#],UnsameQ@@#&]!={}&]],{n,0,15}]
%Y A387328 The strict case is A000009.
%Y A387328 For initial intervals instead of partitions we have A238873, complement A387118.
%Y A387328 For divisors instead of partitions we have A239312, complement A370320.
%Y A387328 For prime factors instead of partitions we have A370592, ranks A368100.
%Y A387328 The complement for prime factors is A370593, ranks A355529.
%Y A387328 The complement is counted by A387134, ranks A387577.
%Y A387328 For sets of strict partitions we have A387178, complement A387137.
%Y A387328 These partitions are ranked by A387576.
%Y A387328 A000005 counts divisors.
%Y A387328 A000041 counts integer partitions.
%Y A387328 Cf. A052335, A052337, A335433, A367771, A370585, A370804.
%K A387328 nonn,new
%O A387328 0,4
%A A387328 _Gus Wiseman_, Sep 01 2025