A362549 Number of partitions of [n] whose blocks can be ordered such that the i-th block (except possibly the last) has at least i elements and no block j > i has an element smaller than the i-th smallest element of block i.

1, 1, 2, 4, 9, 23, 64, 187, 566, 1777, 5820, 19944, 71343, 264719, 1011292, 3953381, 15756609, 63945484, 264384828, 1115246518, 4806957739, 21189601861, 95516470253, 439777682222, 2064164172616, 9853934668051, 47736608806520, 234235866539512, 1162618720397931

Offset: 0

Author

Alois P. Heinz, Apr 24 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) = 9: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234, 1|23|4.
a(5) = 23: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45.
a(6) = 64: 123456, 12345|6, 12346|5, 1234|56, 12356|4, ..., 1|2356|4, 1|235|46, 16|23|45, 1|236|45, 1|23|456.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..888
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A007476, A092920, A210540, A362635, A362637, A362893.

Programs

Maple

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