A300335 Number of ordered set partitions of {1,...,n} with weakly increasing block-sums.

1, 1, 2, 6, 18, 65, 258, 1156, 5558, 29029, 161942, 967921, 6110687, 40807420, 286177944, 2107745450, 16202590638, 130043111849, 1085011337141, 9408577992091, 84501248359552, 786018565954838, 7550153439748394

Offset: 0

Author

Gus Wiseman, Mar 03 2018

Keywords

• nonn
• more

Examples

			The a(3) = 6 ordered set partitions: (123), (1)(23), (2)(13), (12)(3), (3)(12), (1)(2)(3).
		

Crossrefs

A000110(n) <= a(n) <= A000670(n).

Cf. A005651, A063834, A275780, A279375, A279790, A279791, A281113.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Times@@Factorial/@Length/@GatherBy[sptn,Total],{sptn,sps[Range[n]]}],{n,8}]

Extensions

a(12)-a(15) from Alois P. Heinz, Mar 03 2018

a(16)-a(22) from Christian Sievers, Aug 30 2024