A362637 Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i (except possibly the last) has size at least i.

1, 1, 2, 4, 10, 30, 96, 323, 1184, 4784, 20708, 93073, 431004, 2080610, 10615276, 57291063, 322921896, 1871715144, 11065738360, 66843918825, 415837464280, 2684434034706, 18010208402784, 124877499979859, 886741484322660, 6399683149311272, 46802092819866340

Offset: 0

Author

Alois P. Heinz, Apr 28 2023

Keywords

• nonn

Examples

			a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(2) = 2: 12, 1|2.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 10: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234, 1|23|4, 1|24|3.
a(5) = 30: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45, 14|25|3, 15|24|3, 1|245|3, 1|24|35, 1|25|34.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..695
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A362549, A362635, A362639.

Programs

Maple

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