OFFSET
0,4
COMMENTS
An integer set s is gapless if s = {} or |s| = max(s) - min(s) + 1.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..700
Wikipedia, Partition of a set
FORMULA
a(n) = 1 <=> n in { A131577 }.
EXAMPLE
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 1: 1234.
a(5) = 11: 12345, 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234.
a(6) = 61: 123456, 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, 125|36|4, 125|3|46, 126|35|4, 12|356|4, 126|3|45, 12|3|456, 134|25|6, 134|26|5, 134|2|56, 135|24|6, 13|245|6, 136|24|5, 13|246|5, 135|26|4, 135|2|46, 136|25|4, 13|256|4, 136|2|45, 13|2|456, 145|23|6, 14|235|6, 146|23|5, 14|236|5, 15|234|6, 16|234|5, 1|234|56, 156|23|4, 15|236|4, 16|235|4, 1|235|46, 1|236|45, 1|23|456, 145|26|3, 145|2|36, 146|25|3, 14|256|3, 146|2|35, 14|2|356, 156|24|3, 15|246|3, 16|245|3, 1|245|36, 1|246|35, 1|24|356, 156|2|34, 15|2|346, 1|256|34, 1|25|346, 16|2|345, 1|26|345.
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
`if`(i>n, 0, b(n-i, i-1)*binomial(n, i))))
end:
a:= n-> add(b(n, i), i=signum(n)..n):
seq(a(n), n=0..29);
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 02 2026
STATUS
approved
