A363454 Number of partitions of [n] such that the number of blocks containing only odd elements equals the number of blocks containing only even elements and no block contains both odd and even elements.

1, 0, 1, 1, 2, 4, 11, 28, 87, 266, 952, 3381, 13513, 53915, 237113, 1046732, 5016728, 24186664, 125121009, 652084528, 3615047527, 20211789423, 119384499720, 711572380960, 4455637803543, 28162688795697, 186152008588691, 1242276416218540, 8636436319397292

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..650

Wikipedia, Partition of a set

Formula

a(n) = Sum_{k=0..floor(n/2)} Stirling2(floor(n/2),k) * Stirling2(ceiling(n/2),k).

a(2n) = A047797(n).

Example

a(0) = 1: () the empty partition.
a(1) = 0.
a(2) = 1: 1|2.
a(3) = 1: 13|2.
a(4) = 2: 13|24, 1|2|3|4.
a(5) = 4: 135|24, 13|2|4|5, 15|2|3|4, 1|2|35|4.
a(6) = 11: 135|246, 13|24|5|6, 13|26|4|5, 13|2|46|5, 15|24|3|6, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46, 1|2|3|4|5|6.
a(7) = 28: 1357|246, 135|24|6|7, 137|24|5|6, 13|24|57|6, 135|26|4|7, 135|2|46|7, 137|26|4|5, 13|26|4|57, 137|2|46|5, 13|2|46|57, 13|2|4|5|6|7, 157|24|3|6, 15|24|37|6, 17|24|35|6, 1|24|357|6, 157|26|3|4, 15|26|37|4, 157|2|3|46, 15|2|37|46, 15|2|3|4|6|7, 17|26|35|4, 1|26|357|4, 17|2|35|46, 1|2|357|46, 1|2|35|4|6|7, 17|2|3|4|5|6, 1|2|37|4|5|6, 1|2|3|4|57|6.

Maple

a:= n-> (h-> add(Stirling2(h, k)*Stirling2(n-h, k), k=0..h))(iquo(n, 2)):
seq(a(n), n=0..40);
# second Maple program:
b:= proc(n, x, y) option remember; `if`(abs(x-y)>n, 0,
      `if`(n=0, 1, `if`(x>0, b(n-1, y, x)*x, 0)+b(n-1, y, x+1)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..40);

Crossrefs

Bisection gives A047797 (even part).

Cf. A000110, A124419, A124425, A363451.

Sequence in context: A148134 A151425 A148135 * A034768 A148136 A148137

Adjacent sequences: A363451 A363452 A363453 * A363455 A363456 A363457

Keyword

nonn

Author

Alois P. Heinz, Jun 02 2023

Status

approved