A355383 Number of pairs (y, v), where y is a partition of n and v is a sub-multiset of y whose cardinality equals the number of distinct parts in y.

1, 1, 2, 3, 6, 10, 16, 26, 42, 64, 100, 150, 224, 330, 482, 697, 999, 1418, 1996, 2794, 3879, 5355, 7343, 10018, 13583, 18338, 24618, 32917, 43790, 58043, 76591, 100716, 131906, 172194, 223966, 290423, 375318, 483668, 621368, 796138, 1017146

Offset

0,3

Comments

If a partition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of partitions.

Links

Table of n, a(n) for n=0..40.

Example

The a(0) = 1 through a(5) = 10 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)
                (11)(1)  (21)(21)  (31)(31)   (41)(41)
                         (111)(1)  (22)(2)    (32)(32)
                                   (211)(11)  (311)(11)
                                   (211)(21)  (311)(31)
                                   (1111)(1)  (221)(21)
                                              (221)(22)
                                              (2111)(11)
                                              (2111)(21)
                                              (11111)(1)

Mathematica

Table[Sum[Length[Union[Subsets[y, {Length[Union[y]]}]]], {y, IntegerPartitions[n]}], {n, 0, 15}]

Crossrefs

With multiplicity we have A339006.

The version for compositions is A355384.

The homogeneous version w/o containment is A355385, compositions A355388.

A001970 counts multiset partitions of partitions.

A063834 counts partitions of each part of a partition.

Splitting partitions: A072706, A316245, A317715, A318683, A318684, A319794, A323433, A323583, A336131-A336136.

Cf. A000009, A022811, A032020, A070933, A181591, A181819, A319910, A355382.

Sequence in context: A101277 A262984 A201077 * A023655 A354210 A023561

Adjacent sequences: A355380 A355381 A355382 * A355384 A355385 A355386

Keyword

nonn

Author

Gus Wiseman, Jul 02 2022

Status

approved