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

1, 1, 0, 1, 0, 0, 3, 1, 0, 0, 60, 45, 53, 24, 7, 12601, 15120, 33390, 55710, 66522, 86037, 37907754, 63130067, 202203684, 511378789, 1421634137, 2566309603, 5855352202, 2064277450957, 4418631559288, 18485494082571, 61020702809287, 232959438927000, 783244248553960

Offset: 0

Author

Alois P. Heinz, Jul 13 2023

Keywords

• nonn

Comments

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

There are A188431(n) different block size configurations for a given n. - John Tyler Rascoe, Jul 19 2023

Examples

			a(0) = 1: () the empty partition.
a(1) = 1: 1.
a(3) = 1: 1|23.
a(6) = 3: 1|24|356, 1|25|346, 1|26|345.
a(7) = 1: 1|23|4567.
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.
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).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..735
• Wikipedia, Partition of a set

Crossrefs

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

Programs

Maple

b:= proc(n, i) option remember; `if`(i*(i+1)/2n or i>n-i+1, 0, b(n-i, i-1)*binomial(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);  # Alois P. Heinz, Jul 22 2023

Mathematica

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]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz *)