cp's OEIS Frontend

A387330 Number of integer partitions of n such that it is possible to choose a different constant integer partition of each part.

%I A387330 #12 Sep 09 2025 08:12:20
%S A387330 1,1,1,2,3,4,5,7,10,12,16,21,27,34,43,54,67,83,103,126,155,188,229,
%T A387330 277,335,403,483,578,691,821,975,1155,1367,1610,1896,2228,2613,3057,
%U A387330 3573,4167,4853,5640,6550,7590,8786,10154,11722,13510,15556,17885,20540
%N A387330 Number of integer partitions of n such that it is possible to choose a different constant integer partition of each part.
%C A387330 Also the number of integer partitions of n such that for each part k the multiplicity of k is at most A000005(k).
%e A387330 The partition (4,2,2,1) has choices such as ((2,2),(1,1),(2),(1)) so is counted under a(9).
%e A387330 The a(1) = 1 through a(9) = 12 partitions:
%e A387330   (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
%e A387330             (21)  (22)  (32)   (33)   (43)   (44)    (54)
%e A387330                   (31)  (41)   (42)   (52)   (53)    (63)
%e A387330                         (221)  (51)   (61)   (62)    (72)
%e A387330                                (321)  (322)  (71)    (81)
%e A387330                                       (331)  (332)   (432)
%e A387330                                       (421)  (422)   (441)
%e A387330                                              (431)   (522)
%e A387330                                              (521)   (531)
%e A387330                                              (3221)  (621)
%e A387330                                                      (3321)
%e A387330                                                      (4221)
%t A387330 consptns[n_]:=Select[IntegerPartitions[n],SameQ@@#&];
%t A387330 Table[Length[Select[IntegerPartitions[n],Select[Tuples[consptns/@#],UnsameQ@@#&]!={}&]],{n,0,15}]
%Y A387330 For initial intervals instead of constant partitions we have A238873, complement A387118.
%Y A387330 For divisors instead of constant partitions we have A239312, complement A370320.
%Y A387330 The complement for all partitions is A387134, ranks A387577.
%Y A387330 The complement for strict partitions is A387137.
%Y A387330 For strict instead of constant partitions we have A387178.
%Y A387330 These partitions are ranked by A387181.
%Y A387330 For all partitions (not just constant) we have A387328, ranks A387576.
%Y A387330 The complement is counted by A387329, ranks A387180.
%Y A387330 A000005 counts constant integer partitions.
%Y A387330 A000009 counts strict integer partitions.
%Y A387330 A000041 counts integer partitions.
%Y A387330 A063834 counts twice-partitions.
%Y A387330 Cf. A052335, A276079, A355731, A355739, A367771, A383706, A387110.
%K A387330 nonn,new
%O A387330 0,4
%A A387330 _Gus Wiseman_, Sep 07 2025