A364207 - OEIS

%I #28 Oct 21 2023 01:44:49

%S 1,1,0,1,0,0,3,1,0,0,60,45,53,24,7,12601,15120,33390,55710,66522,

%T 86037,37907754,63130067,202203684,511378789,1421634137,2566309603,

%U 5855352202,2064277450957,4418631559288,18485494082571,61020702809287,232959438927000,783244248553960

%N Number of partitions of [n] such that the minimal element of each block is also its size.

%C The block sizes are distinct as a consequence of the definition.

%C There are A188431(n) different block size configurations for a given n. - _John Tyler Rascoe_, Jul 19 2023

%H Alois P. Heinz, <a href="/A364207/b364207.txt">Table of n, a(n) for n = 0..735</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(3) = 1: 1|23.

%e a(6) = 3: 1|24|356, 1|25|346, 1|26|345.

%e a(7) = 1: 1|23|4567.

%e a(10) = 60: 1|25|367|489(10), 1|25|368|479(10), 1|25|369|478(10), ..., 1|28|39(10)|4567, 1|29|38(10)|4567, 1|2(10)|389|4567.

%e a(14) = 7: 1|23|4568|79(10)(11)(12)(13)(14), 1|23|4569|78(10)(11)(12)(13)(14), 1|23|456(10)|789(11)(12)(13)(14), 1|23|456(11)|789(10)(12)(13)(14), 1|23|456(12)|789(10)(11)(13)(14), 1|23|456(13)|789(10)(11)(12)(14), 1|23|456(14)|789(10)(11)(12)(13).

%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,

%p       b(n, i-1)+`if`(i>n or i>n-i+1, 0, b(n-i, i-1)*binomial(n-i, i-1))))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..33);  # _Alois P. Heinz_, Jul 22 2023

%t b[n_, i_] := b[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, 1, b[n, i-1] + If[i > n || i > n-i+1, 0, b[n-i, i-1]*Binomial[n-i, i-1]]]];

%t a[n_] := b[n, n];

%t Table[a[n], {n, 0, 33}] (* _Jean-François Alcover_, Oct 20 2023, after _Alois P. Heinz_ *)

%Y Cf. A000110, A007837, A188431, A326493, A327711, A364406.

%K nonn

%O 0,7

%A _Alois P. Heinz_, Jul 13 2023