cp's OEIS Frontend

A364207 Number of partitions of [n] such that the minimal element of each block is also its size.

%I A364207 #28 Oct 21 2023 01:44:49
%S A364207 1,1,0,1,0,0,3,1,0,0,60,45,53,24,7,12601,15120,33390,55710,66522,
%T A364207 86037,37907754,63130067,202203684,511378789,1421634137,2566309603,
%U A364207 5855352202,2064277450957,4418631559288,18485494082571,61020702809287,232959438927000,783244248553960
%N A364207 Number of partitions of [n] such that the minimal element of each block is also its size.
%C A364207 The block sizes are distinct as a consequence of the definition.
%C A364207 There are A188431(n) different block size configurations for a given n. - _John Tyler Rascoe_, Jul 19 2023
%H A364207 Alois P. Heinz, <a href="/A364207/b364207.txt">Table of n, a(n) for n = 0..735</a>
%H A364207 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A364207 a(0) = 1: () the empty partition.
%e A364207 a(1) = 1: 1.
%e A364207 a(3) = 1: 1|23.
%e A364207 a(6) = 3: 1|24|356, 1|25|346, 1|26|345.
%e A364207 a(7) = 1: 1|23|4567.
%e A364207 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 A364207 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 A364207 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,
%p A364207       b(n, i-1)+`if`(i>n or i>n-i+1, 0, b(n-i, i-1)*binomial(n-i, i-1))))
%p A364207     end:
%p A364207 a:= n-> b(n$2):
%p A364207 seq(a(n), n=0..33);  # _Alois P. Heinz_, Jul 22 2023
%t A364207 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 A364207 a[n_] := b[n, n];
%t A364207 Table[a[n], {n, 0, 33}] (* _Jean-François Alcover_, Oct 20 2023, after _Alois P. Heinz_ *)
%Y A364207 Cf. A000110, A007837, A188431, A326493, A327711, A364406.
%K A364207 nonn
%O A364207 0,7
%A A364207 _Alois P. Heinz_, Jul 13 2023