A321467 Number of factorizations of n! into factors > 1 that can be obtained by taking the block-products of some set partition of {2,3,...,n}.

1, 1, 1, 2, 5, 15, 47, 183, 719, 3329, 14990, 83798, 393864, 2518898

Offset: 0

Author

Gus Wiseman, Nov 11 2018

Keywords

• nonn
• more

Comments

a(n) is the number of factorizations coarser than (2*3*...*n) in the poset of factorizations of n! into factors > 1, ordered by refinement.

Examples

			The a(1) = 1 through a(5) = 15 factorizations:
  ()  (2)  (6)    (24)     (120)
           (2*3)  (3*8)    (2*60)
                  (4*6)    (3*40)
                  (2*12)   (4*30)
                  (2*3*4)  (5*24)
                           (6*20)
                           (8*15)
                           (10*12)
                           (3*5*8)
                           (4*5*6)
                           (2*3*20)
                           (2*4*15)
                           (2*5*12)
                           (3*4*10)
                           (2*3*4*5)
For example, 10*12 = (2*5)*(3*4), so (10*12) is counted under a(5).
		

Crossrefs

Dominated by A000110.

Cf. A001055, A066723, A157612, A242422, A265947, A317141, A317144, A317145, A317534, A321468, A321470, A321471, A321472, A321514.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Union[Sort/@Apply[Times,sps[Range[2,n]],{2}]]],{n,10}]