A372802 Number of partitions of [n] having exactly one block of maximal size and one block of minimal size (and any number of non-extremal blocks).

0, 1, 1, 4, 5, 16, 82, 169, 1381, 4162, 34346, 109099, 1114610, 5041271, 39441963, 269812729, 1972727781, 14983080612, 126099739072, 989666749503, 8839669627570, 79767000198673, 725399587976669, 6979798715649335, 69812296785011890, 703554021895986941

Offset: 0

Author

Alois P. Heinz, May 13 2024

Keywords

• nonn

Comments

Minimal block and maximal block are identical if there is only one block.

Examples

			a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
a(5) = 16: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
a(8) = 1381: 12345678, 1234567|8, 1234568|7, ..., 1|27|38|456, 18|2|37|456, 1|28|37|456.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..577
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A007837, A224219, A372721.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=i, 1, 0)+`if`(i signum(n)+add(binomial(n,i)*b(n-i, i+1), i=1..(n-1)/2):
seq(a(n), n=0..30);