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

0,3

Links

Table of n, a(n) for n=0..22.

Example

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

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}]

Crossrefs

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

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

Sequence in context: A150073 A150074 A349367 * A108066 A052863 A037128

Adjacent sequences: A300332 A300333 A300334 * A300336 A300337 A300338

Keyword

nonn,more

Author

Gus Wiseman, Mar 03 2018

Extensions

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

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

Status

approved