A363592 Number of partitions of [n] such that in each block the smallest element has the same parity as the largest element.

1, 1, 1, 3, 6, 20, 55, 223, 761, 3595, 14532, 77818, 361605, 2155525, 11274781, 73822175, 428004750, 3046519516, 19348533739, 148493347507, 1023481273549, 8412534272415, 62450994058052, 546699337652602, 4343869829492281, 40308548641909593, 340994681344324137

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..150

Wikipedia, Partition of a set

Formula

a(n) mod 2 = A131719(n+1).

Example

a(0) = 1: () the empty partition.
a(1) = 1: 1.
a(2) = 1: 1|2.
a(3) = 3: 123, 13|2, 1|2|3.
a(4) = 6: 123|4, 13|24, 13|2|4, 1|234, 1|24|3, 1|2|3|4.
a(5) = 20: 12345, 1235|4, 123|4|5, 1245|3, 125|3|4, 1345|2, 135|24, 13|24|5, 135|2|4, 13|2|4|5, 15|234, 1|234|5, 145|2|3, 15|24|3, 1|24|35, 1|24|3|5, 1|2|345, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.

Maple

b:= proc(n, x, y, u, v) option remember; `if`(y+u>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, A131719, A361084.

Sequence in context: A148588 A148589 A148590 * A148591 A148592 A148593

Adjacent sequences: A363589 A363590 A363591 * A363593 A363594 A363595

Keyword

nonn

Author

Alois P. Heinz, Jun 10 2023

Status

approved