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A355385 Number of pairs (y, v) of integer partitions of n where the length of v equals the number of distinct parts in y. 6
1, 1, 2, 3, 7, 12, 25, 43, 81, 141, 243, 409, 699, 1132, 1844, 2995, 4744, 7408, 11655, 17839, 27509, 41546, 62879, 93537, 139974, 205547 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also the number of composable pairs of integer partitions of n, where a partition is regarded as an arrow from (number of parts) to (number of distinct parts). Is there a nice choice of a composition operation making this into an associative category?

LINKS

Table of n, a(n) for n=0..25.

EXAMPLE

The a(0) = 1 through a(5) = 10 pairs:

  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)

                (11)(2)  (21)(21)  (31)(31)   (41)(41)

                         (111)(3)  (31)(22)   (41)(32)

                                   (22)(4)    (32)(41)

                                   (211)(31)  (32)(32)

                                   (211)(22)  (311)(41)

                                   (1111)(4)  (311)(32)

                                              (221)(41)

                                              (221)(32)

                                              (2111)(41)

                                              (2111)(32)

                                              (11111)(5)

MATHEMATICA

Table[Length[Select[Tuples[IntegerPartitions[n], 2], Length[Union[#[[1]]]]==Length[#[[2]]]&]], {n, 0, 15}]

CROSSREFS

The inhomogeneous version with containment and multiplicity is A339006.

The inhomogeneous version with containment is A355383.

The inhomogeneous version with containment for compositions is A355384.

The version for compositions is A355388.

A001970 counts multiset partitions of partitions.

A063834 counts partitions of each part of a partition.

A323583 counts splittings of partitions.

Cf. A000009, A022811, A032020, A070933, A072706, A279787, A319910, A336135.

Sequence in context: A108742 A018240 A090596 * A321838 A298897 A054272

Adjacent sequences:  A355382 A355383 A355384 * A355386 A355387 A355388

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jul 02 2022

STATUS

approved

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Last modified September 29 18:39 EDT 2022. Contains 357090 sequences. (Running on oeis4.)