A387178 - OEIS

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

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OFFSET

0,4

COMMENTS

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).

LINKS

Table of n, a(n) for n=0..51.

EXAMPLE

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)

MATHEMATICA

strptns[n_]:=Select[IntegerPartitions[n], UnsameQ@@#&];

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

CROSSREFS

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.

Sequence in context: A135279 A035631 A050046 * A052337 A308858 A192432

Adjacent sequences: A387175 A387176 A387177 * A387179 A387180 A387181

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 02 2025

STATUS

approved