A370320 -id:A370320 - cp's OEIS frontend

A387178 Number of integer partitions of n whose parts have choosable sets of strict integer partitions.

1, 1, 1, 2, 2, 3, 5, 6, 8, 10, 13, 17, 21, 27, 34, 42, 53, 65, 80, 98, 119, 146, 177, 213, 258, 309, 370, 443, 528, 628, 745, 882, 1043, 1229, 1447, 1700, 1993, 2333, 2727, 3182, 3707, 4311, 5008, 5808, 6727, 7782, 8990, 10371, 11952, 13756, 15815, 18161

Offset: 0

Gus Wiseman, Sep 02 2025

First differs from A052337 in having 745 instead of 746.
We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.
a(n) is the number of integer partitions of n such that it is possible to choose a sequence of distinct strict integer partitions of each part.
Also the number of integer partitions of n with no part k whose multiplicity exceeds A000009(k).

			The partition y = (3,3,2) has sets of strict integer partitions ({(2,1),(3)},{(2,1),(3)},{(2)}), and we have the choice ((2,1),(3),(2)) or ((3),(2,1),(2)), so y is counted under a(8).
The a(1) = 1 through a(9) = 10 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                          (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (3,3,1)  (7,1)    (8,1)
                                          (4,2,1)  (3,3,2)  (4,3,2)
                                                   (4,3,1)  (4,4,1)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)
                                                            (3,3,2,1)

For initial intervals instead of strict partitions we have A238873, ranks A387112.
For divisors instead of strict partitions we have A239312, ranks A368110.
The complement for divisors is A370320, ranks A355740.
For prime factors instead of strict partitions we have A370592, ranks A368100.
The complement for prime factors is A370593, ranks A355529.
The complement for initial intervals is A387118, ranks A387113.
The complement for all partitions is A387134, ranks A387577.
The complement is counted by A387137, ranks A387176.
These partitions are ranked by A387177.
For all partitions instead of just strict partitions we have A387328, ranks A387576.
The complement for constant partitions is A387329, ranks A387180.
For constant partitions instead of strict partitions we have A387330, ranks A387181.
A000041 counts integer partitions, strict A000009.
A358914 counts twice-partitions into distinct strict partitions.
A367902 counts choosable set-systems, complement A367903.
Cf. A005703, A052335, A261049, A270995, A276078, A335448, A367867, A367901, A387115.

strptns[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Select[Tuples[strptns/@#],UnsameQ@@#&]!={}&]],{n,0,15}]

A387180 Numbers of which it is not possible to choose a different constant integer partition of each prime index.

Original entry on oeis.org

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204

Offset: 1

Gus Wiseman, Aug 30 2025

First differs from A276079 in having 125.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also numbers n with at least one prime index k such that the multiplicity of prime(k) in the prime factorization of n exceeds the number of divisors of k.

			The prime indices of 60 are {1,1,2,3}, and we have the following 4 choices of constant partitions:
  ((1),(1),(2),(3))
  ((1),(1),(2),(1,1,1))
  ((1),(1),(1,1),(3))
  ((1),(1),(1,1),(1,1,1))
Since none of these is strict, 60 is in the sequence.
The prime indices of 90 are {1,2,2,3}, and we have the following 4 strict choices:
  ((1),(2),(1,1),(3))
  ((1),(2),(1,1),(1,1,1))
  ((1),(1,1),(2),(3))
  ((1),(1,1),(2),(1,1,1))
So 90 is not in the sequence.

For prime factors instead of constant partitions we have A355529, counted by A370593.
For divisors instead of constant partitions we have A355740, counted by A370320.
The complement for prime factors is A368100, counted by A370592.
The complement for divisors is A368110, counted by A239312.
The complement for initial intervals is A387112, counted by A238873.
For initial intervals instead of partitions we have A387113, counted by A387118.
These are the positions of zero in A387120.
For strict instead of constant partitions we have A387176, counted by A387137.
The complement for strict partitions is A387177, counted by A387178.
Twice-partitions of this type are counted by A387179, constant-block case of A296122.
The complement is A387181 (nonzeros of A387120), counted by A387330.
Partitions of this type are counted by A387329.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A355739, A367771, A387111, A387115.
Cf. A000005, A052335, A063834, A276079, A299200, A299201, A335433, A335448, A355731, A383706, A387110.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[#]],UnsameQ@@#&]=={}&]

A387329 Number of integer partitions of n such that it is not possible to choose a different constant integer partition of each part.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 8, 12, 18, 26, 35, 50, 67, 92, 122, 164, 214, 282, 364, 472

Offset: 0

Gus Wiseman, Sep 05 2025

			The a(2) = 1 through a(8) = 12 partitions:
  (11)  (111)  (211)   (311)    (222)     (511)      (611)
               (1111)  (2111)   (411)     (2221)     (2222)
                       (11111)  (2211)    (3211)     (3311)
                                (3111)    (4111)     (4211)
                                (21111)   (22111)    (5111)
                                (111111)  (31111)    (22211)
                                          (211111)   (32111)
                                          (1111111)  (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

For divisors instead of constant partitions we have A370320, complement A239312.
For all (not just constant) partitions we have A387134, ranks A387577.
The complement all partitions is A387328, ranks A387576.
The complement strict partitions is A387178.
For strict (not just constant) partitions we have A387137.
These partitions are ranked by A387180.
The complement is A387330, ranked by A387181.
A000005 counts constant integer partitions.
A000009 counts strict integer partitions.
A000041 counts integer partitions.
Cf. A052335, A063834, A335448, A355739, A387110, A387115.

consptns[n_]:=Select[IntegerPartitions[n],SameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Select[Tuples[consptns/@#],UnsameQ@@#&]=={}&]],{n,0,15}]

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

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 17, 22, 28, 36, 46, 58, 73, 91, 114, 141, 174, 214, 262, 320, 389, 472, 571, 688, 828, 993, 1189, 1419, 1690, 2009, 2383, 2821, 3334, 3931, 4628, 5439, 6381, 7474, 8741, 10207, 11902, 13858, 16114, 18710, 21698, 25130, 29070

Offset: 0

Gus Wiseman, Sep 01 2025

First differs from A052335 at A052335(20) = 173, a(20) = 174, corresponding to the partition (4,4,4,4,4).
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.
Also the number of integer partitions y of n with no part k whose multiplicity in y exceeds A000041(k).

			The a(1) = 1 through a(9) = 13 partitions:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
            (21)  (22)  (32)   (33)   (43)   (44)    (54)
                  (31)  (41)   (42)   (52)   (53)    (63)
                        (221)  (51)   (61)   (62)    (72)
                               (321)  (322)  (71)    (81)
                                      (331)  (332)   (333)
                                      (421)  (422)   (432)
                                             (431)   (441)
                                             (521)   (522)
                                             (3221)  (531)
                                                     (621)
                                                     (3321)
                                                     (4221)

The strict case is A000009.
For initial intervals instead of partitions we have A238873, complement A387118.
For divisors instead of partitions we have A239312, complement A370320.
For prime factors instead of partitions we have A370592, ranks A368100.
The complement for prime factors is A370593, ranks A355529.
The complement is counted by A387134, ranks A387577.
For sets of strict partitions we have A387178, complement A387137.
These partitions are ranked by A387576.
A000005 counts divisors.
A000041 counts integer partitions.
Cf. A052335, A052337, A335433, A367771, A370585, A370804.

Table[Length[Select[IntegerPartitions[n],Select[Tuples[IntegerPartitions/@#],UnsameQ@@#&]!={}&]],{n,0,15}]

cp's OEIS Frontend

A387178 Number of integer partitions of n whose parts have choosable sets of strict integer partitions.

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A387180 Numbers of which it is not possible to choose a different constant integer partition of each prime index.

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A387329 Number of integer partitions of n such that it is not possible to choose a different constant integer partition of each part.

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

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