A362639 - OEIS

%I #9 May 15 2023 16:38:19

%S 1,1,1,1,2,3,4,15,36,70,120,756,2800,7920,18900,40040,388080,2106000,

%T 8408400,27489000,77837760,197520960,2756754000,20903929200,

%U 113809696000,497097881280,1847907341280,6062876820000,17990209036800,343877493960000,3501594297801600

%N Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i (except possibly the last) has size i.

%H Alois P. Heinz, <a href="/A362639/b362639.txt">Table of n, a(n) for n = 0..724</a>

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

%e a(0) = 1: (), the empty partition.

%e a(1) = 1: 1.

%e a(2) = 1: 1|2.

%e a(3) = 1: 1|23.

%e a(4) = 2: 1|23|4, 1|24|3.

%e a(5) = 3: 1|23|45, 1|24|35, 1|25|34.

%e a(6) = 4: 1|23|456, 1|24|356, 1|25|346, 1|26|345.

%e a(7) = 15: 1|23|456|7, 1|23|457|6, 1|23|467|5, 1|24|356|7, 1|24|357|6, 1|24|367|5, 1|25|346|7, 1|25|347|6, 1|26|345|7, 1|27|345|6, 1|26|347|5, 1|27|346|5, 1|25|367|4, 1|26|357|4, 1|27|356|4.

%p b:= proc(n, t) option remember; `if`(n<=t, 1,

%p       b(n-t, t+1)*binomial(n-1, t-1))

%p     end:

%p a:= n-> b(n, 1):

%p seq(a(n), n=0..30);

%Y Cf. A000110, A362637.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Apr 28 2023