A361084 Number of partitions of [n] such that in each block the smallest element and the largest element have opposite parities.

1, 0, 1, 0, 3, 2, 17, 28, 171, 430, 2617, 8496, 54739, 214714, 1477153, 6743204, 49550011, 256645926, 2010328585, 11602635128, 96590823907, 612918061426, 5404902119025, 37319203169580, 347468152001739, 2589081441826334, 25375080898848729, 202668739104752960

Offset: 0

Author

Alois P. Heinz, Jun 12 2023

Keywords

• nonn

Examples

			a(0) = 1: () the empty partition.
a(1) = 0.
a(2) = 1: 12.
a(3) = 0.
a(4) = 3: 1234, 12|34, 14|23.
a(5) = 2: 134|25, 14|235.
a(6) = 17: 123456, 1234|56, 1236|45, 124|356, 1256|34, 12|3456, 12|34|56, 12|36|45, 1346|25, 136|245, 1456|23, 146|235, 14|23|56, 16|2345, 16|23|45, 14|25|36, 16|25|34.
a(7) = 28: 12356|47, 1236|457, 12|356|47, 12|36|457, 13456|27, 1346|257, 134|2567, 134|25|67, 134|27|56, 1356|247, 136|2457, 136|25|47, 136|27|45, 1456|237, 146|2357, 14|23567, 14|235|67, 14|237|56, 156|2347, 16|23457, 156|23|47, 16|235|47, 16|237|45, 16|23|457, 14|257|36, 14|27|356, 156|27|34, 16|257|34.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..165
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A059841, A363592.

Programs

Maple

b:= proc(n, x, y, u, v) option remember; `if`(x+v>n, 0, `if`(n=0, 1,
      `if`(y=0, 0, b(n-1, v, u, y-1, x+1)*y)+b(n-1, v, u, y, x+1)+
      `if`(v=0, 0, b(n-1, v-1, u+1, y, x)*v)+b(n-1, v, u, y, x)*(u+x)))
    end:
a:= n-> b(n, 0$4):
seq(a(n), n=0..30);

Formula

a(n) mod 2 = A059841(n).