A392234 Number of partitions of [n] having a gapless set of block sizes.

1, 1, 2, 5, 11, 37, 147, 583, 2480, 13107, 70943, 402679, 2475090, 16053546, 110386083, 801732558, 6031068685, 47577479779, 392456634139, 3345213666061, 29687368998536, 273157457044902, 2592197433584421, 25421709379594488, 257455974507570833, 2681768165840143028

Offset

0,3

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..576

Wikipedia, Partition of a set

Example

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

Maple

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

Crossrefs

Cf. A000110, A393098, A393144.

Sequence in context: A131581 A195985 A367969 * A056301 A001344 A056302

Adjacent sequences: A392231 A392232 A392233 * A392235 A392236 A392237

Keyword

nonn

Author

Alois P. Heinz, Feb 01 2026

Status

approved