cp's OEIS Frontend

A387134 Number of integer partitions of n whose parts do not have choosable sets of integer partitions.

%I A387134 #7 Aug 31 2025 10:45:26
%S A387134 0,0,1,1,2,3,6,8,12,17,25,34,49,65,89,118,158,206,271,349,453,578,740,
%T A387134 935,1186,1486,1865,2322,2890,3572,4415,5423,6659,8134,9927,12062,
%U A387134 14643,17706,21387,25746,30957,37109,44433,53054,63273,75276,89444,106044
%N A387134 Number of integer partitions of n whose parts do not have choosable sets of integer partitions.
%C A387134 Number of integer partitions of n such that it is not possible to choose a sequence of distinct integer partitions, one of each part.
%C A387134 Also the number of integer partitions of n with at least one part k satisfying that the multiplicity of k exceeds the number of integer partitions of k.
%e A387134 The a(2) = 1 through a(8) = 12 partitions:
%e A387134   (11)  (111)  (211)   (311)    (222)     (511)      (611)
%e A387134                (1111)  (2111)   (411)     (2221)     (2222)
%e A387134                        (11111)  (2211)    (3211)     (3311)
%e A387134                                 (3111)    (4111)     (4211)
%e A387134                                 (21111)   (22111)    (5111)
%e A387134                                 (111111)  (31111)    (22211)
%e A387134                                           (211111)   (32111)
%e A387134                                           (1111111)  (41111)
%e A387134                                                      (221111)
%e A387134                                                      (311111)
%e A387134                                                      (2111111)
%e A387134                                                      (11111111)
%t A387134 Table[Length[Select[IntegerPartitions[n],Length[Select[Tuples[IntegerPartitions/@#],UnsameQ@@#&]]==0&]],{n,0,15}]
%Y A387134 These partitions are ranked by A276079.
%Y A387134 For divisors instead of partitions we have A370320, complement A239312.
%Y A387134 The complement for prime factors is A370592, ranks A368100.
%Y A387134 For prime factors instead of partitions we have A370593, ranks A355529.
%Y A387134 For initial intervals instead of partitions we have A387118, complement A238873.
%Y A387134 For just choices of strict partitions we have A387137.
%Y A387134 The complement is counted by A387328, ranks A276078.
%Y A387134 A000005 counts divisors.
%Y A387134 A000041 counts integer partitions, strict A000009.
%Y A387134 Cf. A335433, A355535, A367867, A367901, A367903, A367905, A367907, A370583, A370594.
%K A387134 nonn,new
%O A387134 0,5
%A A387134 _Gus Wiseman_, Aug 29 2025