A362635 Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i has size at least i.

1, 1, 1, 2, 5, 12, 31, 97, 351, 1318, 4963, 19391, 82531, 386704, 1926907, 9811733, 50252175, 261462430, 1415025895, 8118274255, 49355434511, 312266428040, 2012834117143, 13055850467371, 85215848844559, 565353777291346, 3866868949795579, 27548261709035105

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..700

Wikipedia, Partition of a set

Example

a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 123, 1|23.
a(4) = 5: 1234, 12|34, 13|24, 14|23, 1|234.
a(5) = 12: 12345, 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
a(6) = 31: 123456, 1234|56, 1235|46, 1236|45, 123|456, 1245|36, 1246|35, 124|356, 1256|34, 125|346, 126|345, 12|3456, 1345|26, 1346|25, 134|256, 1356|24, 135|246, 136|245, 13|2456, 1456|23, 145|236, 146|235, 14|2356, 156|234, 15|2346, 16|2345, 1|23456, 1|23|456, 1|24|356, 1|25|346, 1|26|345.

Maple

b:= proc(n, t) option remember; `if`(n=0, 1, add(
      b(n-j, t+1)*binomial(n-1, j-1), j=t..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);

Crossrefs

Cf. A000110, A362549, A362637, A362638, A362893.

Sequence in context: A090826 A132441 A000840 * A232215 A265265 A293868

Adjacent sequences: A362632 A362633 A362634 * A362636 A362637 A362638

Keyword

nonn

Author

Alois P. Heinz, Apr 28 2023

Status

approved