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

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..165

Wikipedia, Partition of a set

Formula

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

Example

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.

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);

Crossrefs

Cf. A000110, A059841, A363592.

Sequence in context: A189731 A342140 A126354 * A158939 A173795 A249769

Adjacent sequences: A361081 A361082 A361083 * A361085 A361086 A361087

Keyword

nonn

Author

Alois P. Heinz, Jun 12 2023

Status

approved