A336342 Number of ways to choose a partition of each part of a strict composition of n.

1, 1, 2, 7, 11, 29, 81, 155, 312, 708, 1950, 3384, 7729, 14929, 32407, 81708, 151429, 305899, 623713, 1234736, 2463743, 6208978, 10732222, 22487671, 43000345, 86573952, 160595426, 324990308, 744946690, 1336552491, 2629260284, 5050032692, 9681365777

Offset

0,3

Comments

A strict composition of n is a finite sequence of distinct positive integers summing to n.

Is there a simple generating function?

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

G.f.: Sum_{k>=0} k! * [y^k](Product_{j>=1} 1 + y*x^j*A000041(j)). - Andrew Howroyd, Apr 16 2021

Example

The a(1) = 1 through a(4) = 11 ways:
  (1)  (2)    (3)        (4)
       (1,1)  (2,1)      (2,2)
              (1,1,1)    (3,1)
              (1),(2)    (1),(3)
              (2),(1)    (2,1,1)
              (1),(1,1)  (3),(1)
              (1,1),(1)  (1,1,1,1)
                         (1),(2,1)
                         (2,1),(1)
                         (1),(1,1,1)
                         (1,1,1),(1)

Mathematica

Table[Length[Join@@Table[Tuples[IntegerPartitions/@ctn], {ctn, Join@@Permutations/@Select[IntegerPartitions[n], UnsameQ@@#&]}]], {n, 0, 10}]

Prog

(PARI) seq(n)={[subst(serlaplace(p), y, 1) | p<-Vec(prod(k=1, n, 1 + y*x^k*numbpart(k) + O(x*x^n)))]} \\ Andrew Howroyd, Apr 16 2021

Crossrefs

Multiset partitions of partitions are A001970.

Strict compositions are counted by A032020, A072574, and A072575.

Splittings of partitions are A323583.

Splittings of partitions with distinct sums are A336131.

Cf. A008289, A011782, A304786, A316245, A317715, A336128, A336132, A336134, A336135.

Partitions:

- Partitions of each part of a partition are A063834.

- Compositions of each part of a partition are A075900.

- Strict partitions of each part of a partition are A270995.

- Strict compositions of each part of a partition are A336141.

Strict partitions:

- Partitions of each part of a strict partition are A271619.

- Compositions of each part of a strict partition are A304961.

- Strict partitions of each part of a strict partition are A279785.

- Strict compositions of each part of a strict partition are A336142.

Compositions:

- Partitions of each part of a composition are A055887.

- Compositions of each part of a composition are A133494.

- Strict partitions of each part of a composition are A304969.

- Strict compositions of each part of a composition are A307068.

Strict compositions:

- Partitions of each part of a strict composition are A336342.

- Compositions of each part of a strict composition are A336127.

- Strict partitions of each part of a strict composition are A336343.

- Strict compositions of each part of a strict composition are A336139.

Sequence in context: A055469 A361151 A327552 * A284354 A228076 A123151

Adjacent sequences: A336339 A336340 A336341 * A336343 A336344 A336345

Keyword

nonn

Author

Gus Wiseman, Jul 18 2020

Status

approved