A392234 - OEIS

%I #30 Feb 03 2026 16:52:47

%S 1,1,2,5,11,37,147,583,2480,13107,70943,402679,2475090,16053546,

%T 110386083,801732558,6031068685,47577479779,392456634139,

%U 3345213666061,29687368998536,273157457044902,2592197433584421,25421709379594488,257455974507570833,2681768165840143028

%N Number of partitions of [n] having a gapless set of block sizes.

%C An integer set s is gapless if s = {} or |s| = max(s) - min(s) + 1.

%H Alois P. Heinz, <a href="/A392234/b392234.txt">Table of n, a(n) for n = 0..576</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e a(3) = 5: 123, 12|3, 13|2, 1|23, 1|2|3.

%e a(4) = 11: 1234, 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.

%e a(5) = 37: 12345, 123|45, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 12|3|4|5, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 13|2|4|5, 145|23, 14|235, 14|23|5, 15|234, 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.

%p b:= proc(n, i) option remember; `if`(n=0 or i=1, 1, add(1/j!*

%p       b(n-i*j, i-1)*combinat[multinomial](n, n-i*j, i$j), j=1..n/i))

%p     end:

%p a:= n-> add(b(n, i), i=signum(n)..n):

%p seq(a(n), n=0..25);

%Y Cf. A000110, A393098, A393144.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Feb 01 2026