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

Author

Antti Karttunen, Aug 10 2018

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Index entries for sequences related to partitions

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.

Programs

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 *)

PARI

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

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

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