A393098 Number of partitions of [n] whose set of block sizes equals [k] for some k.

1, 1, 1, 4, 7, 26, 121, 442, 2059, 10180, 61471, 329066, 2138137, 13584364, 95688685, 696565066, 5282656591, 41857905632, 347337106399, 2975647624930, 26483128785361, 245273192449636, 2329402934700997, 22981837755044114, 232971879903054415, 2438618971957716376

Offset

0,4

Comments

Number of partitions of [n] whose set of block sizes s satisfies: s = {} or |s| = max(s).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..576

Wikipedia, Partition of a set

Formula

a(n) <= A393037(n) with equality only for n <= 5.

Example

a(5) = 26: 12|34|5, 12|35|4, 12|3|45, 12|3|4|5, 13|24|5, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 15|23|4, 1|23|45, 1|23|4|5, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
a(6) = 121: 123|45|6, 123|46|5, 123|4|56, 124|35|6, 124|36|5, 124|3|56, 125|34|6, 12|345|6, 126|34|5, 12|346|5, 12|34|5|6, 125|36|4, 125|3|46, 126|35|4, 12|356|4, 12|35|4|6, 126|3|45, 12|3|456, 12|3|45|6, 12|36|4|5, 12|3|46|5, 12|3|4|56, 12|3|4|5|6, 134|25|6, 134|26|5, 134|2|56, 135|24|6, 13|245|6, 136|24|5, 13|246|5, 13|24|5|6, 135|26|4, 135|2|46, 136|25|4, 13|256|4, 13|25|4|6, 136|2|45, 13|2|456, 13|2|45|6, 13|26|4|5, 13|2|46|5, 13|2|4|56, 13|2|4|5|6, 145|23|6, 14|235|6, 146|23|5, 14|236|5, 14|23|5|6, 15|234|6, 16|234|5, 1|234|56, 156|23|4, 15|236|4, 15|23|4|6, 16|235|4, 1|235|46, 1|236|45, 1|23|456, 1|23|45|6, 16|23|4|5, 1|23|46|5, 1|23|4|56, 1|23|4|5|6, 145|26|3, 145|2|36, 146|25|3, 14|256|3, 14|25|3|6, 146|2|35, 14|2|356, 14|2|35|6, 14|26|3|5, 14|2|36|5, 14|2|3|56, 14|2|3|5|6, 156|24|3, 15|246|3, 15|24|3|6, 16|245|3, 1|245|36, 1|246|35, 1|24|356, 1|24|35|6, 16|24|3|5, 1|24|36|5, 1|24|3|56, 1|24|3|5|6, 156|2|34, 15|2|346, 15|2|34|6, 1|256|34, 1|25|346, 1|25|34|6, 16|2|345, 1|26|345, 16|2|34|5, 1|26|34|5, 1|2|34|56, 1|2|34|5|6, 15|26|3|4, 15|2|36|4, 15|2|3|46, 15|2|3|4|6, 16|25|3|4, 1|25|36|4, 1|25|3|46, 1|25|3|4|6, 16|2|35|4, 1|26|35|4, 1|2|35|46, 1|2|35|4|6, 16|2|3|45, 1|26|3|45, 1|2|36|45, 1|2|3|45|6, 16|2|3|4|5, 1|26|3|4|5, 1|2|36|4|5, 1|2|3|46|5, 1|2|3|4|56, 1|2|3|4|5|6.

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-i*j, i+1)*
      combinat[multinomial](n, n-i*j, i$j)/j!, j=1..n/i))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..25);

Crossrefs

Cf. A000110, A007837, A022915, A392234, A393037, A393144.

Row sums of A387082.

Sequence in context: A378100 A269744 A303844 * A393037 A197789 A076148

Adjacent sequences: A393095 A393096 A393097 * A393099 A393100 A393101

Keyword

nonn

Author

Alois P. Heinz, Feb 01 2026

Status

approved