A387576 -id:A387576 - cp's OEIS frontend

A387137 Number of integer partitions of n whose parts do not have choosable sets of strict integer partitions.

0, 0, 1, 1, 3, 4, 6, 9, 14, 20, 29, 39, 56, 74, 101, 134, 178, 232, 305, 392, 508, 646, 825, 1042, 1317, 1649, 2066, 2567, 3190, 3937, 4859, 5960, 7306, 8914, 10863, 13183, 15984, 19304, 23288, 28003, 33631, 40272, 48166, 57453, 68448, 81352, 96568, 114383

Offset: 0

Gus Wiseman, Sep 02 2025

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 not possible to choose a sequence of distinct strict integer partitions, one of each part.
Also the number of integer partitions of n with at least one part k whose multiplicity exceeds A000009(k).

			The a(2) = 1 through a(8) = 14 partitions:
  (11)  (111)  (22)    (221)    (222)     (322)      (422)
               (211)   (311)    (411)     (511)      (611)
               (1111)  (2111)   (2211)    (2221)     (2222)
                       (11111)  (3111)    (3211)     (3221)
                                (21111)   (4111)     (3311)
                                (111111)  (22111)    (4211)
                                          (31111)    (5111)
                                          (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

The complement for initial intervals is A238873, ranks A387112.
The complement for divisors is A239312, ranks A368110.
Twice-partitions of this type (into distinct strict partitions) are counted by A358914.
For divisors instead of strict partitions we have A370320, ranks A355740.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of strict partitions we have A370593, ranks A355529.
For initial intervals instead of strict partitions we have A387118, ranks A387113.
For all partitions instead of strict partitions we have A387134, ranks A387577.
These partitions are ranked by A387176.
The complement is counted by A387178, ranks A387177.
The complement for partitions is A387328, ranks A387576.
The version for constant partitions is A387329, ranks A387180.
The complement for constant partitions is A387330, ranks A387181.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A367902 counts choosable set-systems, complement A367903.
Cf. A005703, A052335, A261049, A270995, A276078, A335448, A355535, A367867, A367901, A367905, A383706, A387115.

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

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

Original entry on oeis.org

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}]

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}]

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

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 12, 16, 21, 27, 34, 43, 54, 67, 83, 103, 126, 155, 188, 229, 277, 335, 403, 483, 578, 691, 821, 975, 1155, 1367, 1610, 1896, 2228, 2613, 3057, 3573, 4167, 4853, 5640, 6550, 7590, 8786, 10154, 11722, 13510, 15556, 17885, 20540

Offset: 0

Gus Wiseman, Sep 07 2025

Also the number of integer partitions of n such that for each part k the multiplicity of k is at most A000005(k).

			The partition (4,2,2,1) has choices such as ((2,2),(1,1),(2),(1)) so is counted under a(9).
The a(1) = 1 through a(9) = 12 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)   (432)
                                      (421)  (422)   (441)
                                             (431)   (522)
                                             (521)   (531)
                                             (3221)  (621)
                                                     (3321)
                                                     (4221)

For initial intervals instead of constant partitions we have A238873, complement A387118.
For divisors instead of constant partitions we have A239312, complement A370320.
The complement for all partitions is A387134, ranks A387577.
The complement for strict partitions is A387137.
For strict instead of constant partitions we have A387178.
These partitions are ranked by A387181.
For all partitions (not just constant) we have A387328, ranks A387576.
The complement is counted by A387329, ranks A387180.
A000005 counts constant integer partitions.
A000009 counts strict integer partitions.
A000041 counts integer partitions.
A063834 counts twice-partitions.
Cf. A052335, A276079, A355731, A355739, A367771, A383706, A387110.

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

A387326 Numbers whose prime factors do not have choosable sets of integer partitions.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 81

Offset: 1

Gus Wiseman, Sep 04 2025

We say that a set or 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.

Also numbers n with at least one prime index k such that the multiplicity of k in the prime factors of n exceeds the number of integer partitions of k.

			The prime factors of 72 are {2,2,2,3,3}, with sets of partitions ({(1,1),(2)},{(1,1),(2)},{(1,1),(2)},{(1,1),(2)},{(1,1,1),(2,1),(3)},{(1,1,1),(2,1),(3)}), which is not choosable, so 72 is in the sequence.

The version for prime indices differs from A276079 in lacking 16807, counted by A387134.
If we take the set {1..k} instead of the set of integer partitions of k we get A325127.
A subset of A365886.
Positions of zero in A387133.
For prime indices instead of factors we have A387577.
A000041 counts integer partitions, strict A000009.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A387327 counts partitions of prime factors.
A387328 counts partitions with choosable sets of partitions, ranks A387576.
Cf. A063834, A120383, A289509, A299200, A335433, A335448, A355739, A383706, A387111, A387135, A387180.

Select[Range[50],Select[Tuples[IntegerPartitions/@Join@@ConstantArray@@@FactorInteger[#]],UnsameQ@@#&]=={}&]

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

A387137 Number of integer partitions of n whose parts do not have choosable sets of strict integer partitions.

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

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

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A387326 Numbers whose prime factors do not have choosable sets of integer partitions.

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

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