login
Number of set partitions of multiset {1, 2, 2, 3, 3, 3, ..., n X n}.
23

%I #30 Aug 31 2020 19:49:40

%S 1,1,4,52,2776,695541,927908528,7303437156115,371421772559819369,

%T 132348505150329265211927,355539706668772869353964510735,

%U 7698296698535929906799439134946965681,1428662247641961794158621629098030994429958386,2405509035205023556420199819453960482395657232596725626

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

%C Number of factorizations of the superprimorial A006939(n) into factors > 1. - _Gus Wiseman_, Aug 21 2020

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

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

%F a(n) = A001055(A006939(n)). - _Gus Wiseman_, Aug 21 2020

%F a(n) = A318284(A002110(n)). - _Andrew Howroyd_, Aug 31 2020

%e 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.

%p g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+

%p `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,

%p g(n/d, d)), d=divisors(n) minus {1, n}))

%p end:

%p a:= n-> g(mul(ithprime(i)^i, i=1..n)$2):

%p seq(a(n), n=0..5); # _Alois P. Heinz_, Jul 26 2020

%t chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[facs[chern[n]]],{n,3}] (* _Gus Wiseman_, Aug 21 2020 *)

%o (PARI) \\ See A318284 for count.

%o a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ _Andrew Howroyd_, Aug 31 2020

%Y Cf. A033312, A317826.

%Y Subsequence of A317828.

%Y A000142 counts submultisets of the same multiset.

%Y A022915 counts permutations of the same multiset.

%Y A337069 is the strict case.

%Y A001055 counts factorizations.

%Y A006939 lists superprimorials or Chernoff numbers.

%Y A076716 counts factorizations of factorials.

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

%Y A181818 lists products of superprimorials, with complement A336426.

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

%K nonn

%O 0,3

%A _Antti Karttunen_, Aug 10 2018

%E a(0)=1 prepended and a(7) added by _Alois P. Heinz_, Jul 26 2020

%E a(8)-a(13) from _Andrew Howroyd_, Aug 31 2020