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

Author

Alois P. Heinz, Jun 02 2023

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..650
• Wikipedia, Partition of a set

Crossrefs

Bisection gives A047797 (even part).

Cf. A000110, A124419, A124425, A363451.

Programs

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);

Formula

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

a(2n) = A047797(n).