A362638 Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i has size at most i.

1, 1, 1, 2, 4, 11, 34, 117, 458, 1960, 9113, 45730, 246220, 1411820, 8584314, 55121249, 372445462, 2639750490, 19570232608, 151390999949, 1219308754065, 10204045406690, 88571532840007, 796100505025761, 7398510884959638, 70995106473861076, 702542384597885615

Offset: 0

Author

Alois P. Heinz, Apr 28 2023

Keywords

• nonn

Examples

			a(0) = 1: ().
a(1) = 1: 1.
a(2) = 1: 1|2.
a(3) = 2: 1|23, 1|2|3.
a(4) = 4: 1|23|4, 1|24|3, 1|2|34, 1|2|3|4.
a(5) = 11: 1|23|45, 1|23|4|5, 1|24|35, 1|24|3|5, 1|25|34, 1|2|345, 1|2|34|5, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
a(6) = 34: 1|23|456, 1|23|45|6, 1|23|46|5, 1|23|4|56, 1|23|4|5|6, 1|24|356, 1|24|35|6, 1|24|36|5, 1|24|3|56, 1|24|3|5|6, 1|25|346, 1|25|34|6, 1|26|345, 1|2|345|6, 1|26|34|5, 1|2|346|5, 1|2|34|56, 1|2|34|5|6, 1|25|36|4, 1|25|3|46, 1|25|3|4|6, 1|26|35|4, 1|2|356|4, 1|2|35|46, 1|2|35|4|6, 1|26|3|45, 1|2|36|45, 1|2|3|456, 1|2|3|45|6, 1|26|3|4|5, 1|2|36|4|5, 1|2|3|46|5, 1|2|3|4|56, 1|2|3|4|5|6.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..578
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A362635.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1, add(
      b(n-j, t+1)*binomial(n-1, j-1), j=1..min(t, n)))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);