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

0,5

Comments

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).

Links

Table of n, a(n) for n=0..47.

Example

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)

Mathematica

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

Crossrefs

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.

Sequence in context: A227561 A352095 A030712 * A025000 A147790 A048577

Adjacent sequences: A387134 A387135 A387136 * A387138 A387139 A387140

Keyword

nonn

Author

Gus Wiseman, Sep 02 2025

Status

approved