A317829 Number of set partitions of multiset {1, 2, 2, 3, 3, 3, ..., n X n}.

1, 1, 4, 52, 2776, 695541, 927908528, 7303437156115, 371421772559819369, 132348505150329265211927, 355539706668772869353964510735, 7698296698535929906799439134946965681, 1428662247641961794158621629098030994429958386, 2405509035205023556420199819453960482395657232596725626

Offset

0,3

Comments

Number of factorizations of the superprimorial A006939(n) into factors > 1. - Gus Wiseman, Aug 21 2020

Links

Table of n, a(n) for n=0..13.

Index entries for sequences related to partitions

Formula

a(n) = A317826(A033312(n+1)) = A317826((n+1)!-1) = A001055(A076954(n)).

a(n) = A001055(A006939(n)). - Gus Wiseman, Aug 21 2020

a(n) = A318284(A002110(n)). - Andrew Howroyd, Aug 31 2020

Example

For n = 2 we have a multiset {1, 2, 2} which can be partitioned as {{1}, {2}, {2}} or {{1, 2}, {2}} or {{1}, {2, 2}} or {{1, 2, 2}}, thus a(2) = 4.

Maple

g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
     `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
        g(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> g(mul(ithprime(i)^i, i=1..n)$2):
seq(a(n), n=0..5);  # Alois P. Heinz, Jul 26 2020

Mathematica

chern[n_]:=Product[Prime[i]^(n-i+1), {i, n}];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[facs[chern[n]]], {n, 3}] (* Gus Wiseman, Aug 21 2020 *)

Prog

(PARI) \\ See A318284 for count.
a(n) = {if(n==0, 1, count(vector(n, i, i)))} \\ Andrew Howroyd, Aug 31 2020

Crossrefs

Cf. A033312, A317826.

Subsequence of A317828.

A000142 counts submultisets of the same multiset.

A022915 counts permutations of the same multiset.

A337069 is the strict case.

A001055 counts factorizations.

A006939 lists superprimorials or Chernoff numbers.

A076716 counts factorizations of factorials.

A076954 can be used instead of A006939 (cf. A307895, A325337).

A181818 lists products of superprimorials, with complement A336426.

Cf. A000178, A002110, A022559, A027423, A124010, A303279, A318284, A322583, A336417, A336496.

Sequence in context: A009671 A015001 A355612 * A327234 A327373 A193914

Adjacent sequences: A317826 A317827 A317828 * A317830 A317831 A317832

Keyword

nonn

Author

Antti Karttunen, Aug 10 2018

Extensions

a(0)=1 prepended and a(7) added by Alois P. Heinz, Jul 26 2020

a(8)-a(13) from Andrew Howroyd, Aug 31 2020

Status

approved