A374321 Number of partitions of [n] such that the number of blocks of size k is zero or equals k for every k.

1, 1, 0, 0, 3, 15, 0, 0, 0, 280, 2800, 0, 0, 600600, 8408400, 0, 2627625, 44669625, 0, 0, 38192529375, 802043116875, 0, 0, 0, 1508282884484376, 39215354996593776, 0, 0, 107469680368165243128, 3224090411044957293840, 0, 0, 0, 76290792475347121351680

Offset: 0

Author

Alois P. Heinz, Jul 04 2024

Keywords

• nonn

Examples

			a(0) = 1: the empty partition.
a(1) = 1: 1.
a(4) = 3: 12|34, 13|24, 14|23.
a(5) = 15: 12|34|5, 12|35|4, 12|3|45, 13|24|5, 13|25|4, 13|2|45, 14|23|5, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
a(9) = 280: 123|456|789, 123|457|689, 123|458|679, 123|459|678, ..., 169|278|345,  178|269|345, 179|268|345, 189|267|345.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..607
• Wikipedia, Partition of a set

Crossrefs

Cf. A001422, A003995, A372579, A374319, A374320, A374329.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, add(`if`(j=0 or j=i, b(n-i*j, i-1)/j!*
      combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);

Formula

a(n) = 0 <=> n in { A001422 }.

a(n) > 0 <=> n in { A003995 }.