A387118 Number of integer partitions of n without choosable initial intervals.

0, 0, 1, 1, 2, 4, 6, 8, 13, 19, 28, 37, 52, 70, 97, 130, 172, 224, 293, 378, 492, 630, 806, 1018, 1286, 1609, 2019, 2514, 3131, 3874, 4784, 5872, 7198, 8786, 10712, 13013, 15794, 19100, 23063, 27752, 33341, 39939, 47781, 57013, 67955, 80816, 95992, 113773, 134668

Offset

0,5

Comments

The initial interval of a nonnegative integer x is the set {1,...,x}.

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.

Conjecture: Also the number of non-superdiagonal reversed integer partitions of n. - Gus Wiseman, Oct 04 2025

Links

Table of n, a(n) for n=0..48.

Example

The partition y = (2,2,1) has initial intervals ({1,2},{1,2},{1}), which are not choosable, so y is counted under a(5).
The a(2) = 1 through a(8) = 13 partitions:
  (11)  (111)  (211)   (221)    (222)     (511)      (611)
               (1111)  (311)    (411)     (2221)     (2222)
                       (2111)   (2211)    (3211)     (3221)
                       (11111)  (3111)    (4111)     (3311)
                                (21111)   (22111)    (4211)
                                (111111)  (31111)    (5111)
                                          (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Select[Tuples[Range/@#], UnsameQ@@#&]=={}&]], {n, 0, 10}]

Crossrefs

The complement is counted by A238873, ranks A387112.

The complement for divisors is A239312, ranks A368110.

For divisors instead of initial intervals we have A370320, ranks A355740.

The complement for prime factors is A370592, ranks A368100.

For prime factors instead of initial intervals we have A370593, ranks A355529.

These partitions have ranks A387113.

For partitions instead of initial intervals we have A387134.

For strict partitions instead of initial intervals we have A387137, ranks A387176.

The complement for strict partitions is A387178.

The complement for partitions is A387328.

Dominates A387578.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A367902 counts choosable set-systems, complement A367903.

A370582 counts sets with choosable prime factors, complement A370583.

Cf. A005703, A052335, A367867, A367901, A367905, A367907, A370585, A370594, A387111.

Sequence in context: A039846 A367218 A094092 * A072791 A058320 A014320

Adjacent sequences: A387115 A387116 A387117 * A387119 A387120 A387121

Keyword

nonn

Author

Gus Wiseman, Sep 02 2025

Extensions

More terms from Jinyuan Wang, Sep 05 2025

Status

approved