A362635 Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i has size at least i.

1, 1, 1, 2, 5, 12, 31, 97, 351, 1318, 4963, 19391, 82531, 386704, 1926907, 9811733, 50252175, 261462430, 1415025895, 8118274255, 49355434511, 312266428040, 2012834117143, 13055850467371, 85215848844559, 565353777291346, 3866868949795579, 27548261709035105

Offset: 0

Author

Alois P. Heinz, Apr 28 2023

Keywords

• nonn

Examples

			a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 123, 1|23.
a(4) = 5: 1234, 12|34, 13|24, 14|23, 1|234.
a(5) = 12: 12345, 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
a(6) = 31: 123456, 1234|56, 1235|46, 1236|45, 123|456, 1245|36, 1246|35, 124|356, 1256|34, 125|346, 126|345, 12|3456, 1345|26, 1346|25, 134|256, 1356|24, 135|246, 136|245, 13|2456, 1456|23, 145|236, 146|235, 14|2356, 156|234, 15|2346, 16|2345, 1|23456, 1|23|456, 1|24|356, 1|25|346, 1|26|345.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..700
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A362549, A362637, A362638, A362893.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 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);