A367867 -id:A367867 - cp's OEIS frontend

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

0, 0, 1, 1, 2, 3, 6, 8, 12, 17, 25, 34, 49, 65, 89, 118, 158, 206, 271, 349, 453, 578, 740, 935, 1186, 1486, 1865, 2322, 2890, 3572, 4415, 5423, 6659, 8134, 9927, 12062, 14643, 17706, 21387, 25746, 30957, 37109, 44433, 53054, 63273, 75276, 89444, 106044

Offset: 0

Gus Wiseman, Aug 29 2025

Number of integer partitions of n such that it is not possible to choose a sequence of distinct integer partitions, one of each part.

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.

			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)

These partitions are ranked by A276079.
For divisors instead of partitions we have A370320, complement A239312.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of partitions we have A370593, ranks A355529.
For initial intervals instead of partitions we have A387118, complement A238873.
For just choices of strict partitions we have A387137.
The complement is counted by A387328, ranks A276078.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
Cf. A335433, A355535, A367867, A367901, A367903, A367905, A367907, A370583, A370594.

Table[Length[Select[IntegerPartitions[n],Length[Select[Tuples[IntegerPartitions/@#],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}]

A368532 Minimal numbers whose binary indices of binary indices contradict a strict version of the axiom of choice.

Original entry on oeis.org

7, 25, 30, 42, 45, 51, 53, 54, 60, 75, 77, 78, 83, 85, 86, 90, 92, 99, 101, 102, 105, 108, 113, 114, 116, 120, 385, 390, 408, 428, 434, 436, 458, 460, 466, 468, 482, 484, 488, 496, 642, 645, 668, 680, 689, 692, 713, 716, 721, 724, 728, 737, 740, 752, 771, 773

Offset: 1

Gus Wiseman, Dec 29 2023

nonn

Minimality is relative to the ordering where x < y means the binary indices of x are a subset of those of y (a Boolean algebra).
A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion.
The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The terms the corresponding set-systems begin:
   7: {{1},{2},{1,2}}
  25: {{1},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  42: {{2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  60: {{1,2},{3},{1,3},{2,3}}
  75: {{1},{2},{3},{1,2,3}}
  77: {{1},{1,2},{3},{1,2,3}}
  78: {{2},{1,2},{3},{1,2,3}}
  83: {{1},{2},{1,3},{1,2,3}}
  85: {{1},{1,2},{1,3},{1,2,3}}
  86: {{2},{1,2},{1,3},{1,2,3}}
  90: {{2},{3},{1,3},{1,2,3}}
  92: {{1,2},{3},{1,3},{1,2,3}}
  99: {{1},{2},{2,3},{1,2,3}}

The version for MM-numbers of multiset partitions is A368187.
A000110 counts set partitions.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A007716, A134964, A355529, A367905, A367907.
Cf. A140637, A367867, A367903, A368094, A368097, A368413.

vmin[y_]:=Select[y,Function[s,Select[DeleteCases[y,s], SubsetQ[bpe[s],bpe[#]]&]=={}]];
Select[Range[100],Select[Tuples[bpe/@bpe[#]] ,UnsameQ@@#&]=={}&]//vmin

A370818 Number of sets of nonempty subsets of {1..n} with only one possible way to choose a set of different vertices of each edge.

Original entry on oeis.org

1, 2, 6, 45, 1352, 157647, 63380093, 85147722812, 385321270991130

Offset: 0

Gus Wiseman, Mar 12 2024

			The set-system {{2},{1,2},{2,4},{1,3,4}} has unique choice (2,1,4,3) so is counted under a(4).

This is the unique version of A367902, complement A367903.
Choosing a sequence gives A367904, ranks A367908.
The maximal case is A368601, complement A368600.
This is the restriction of A370638 to A000225.
Factorizations of this type are counted by A370645.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A133686, A134964, A367772, A367867, A368101, A368109, A370584.

Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Union[Sort/@Select[Tuples[#],UnsameQ@@#&]]]==1&]],{n,0,3}]

a(n) = A370638(2^n - 1).

Binomial transform of A368601. - Christian Sievers, Aug 12 2024

a(5)-a(8) from Christian Sievers, Aug 12 2024

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A387134 Number of integer partitions of n whose parts do not have choosable sets of 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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A368532 Minimal numbers whose binary indices of binary indices contradict a strict version of the axiom of choice.

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A370818 Number of sets of nonempty subsets of {1..n} with only one possible way to choose a set of different vertices of each edge.

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