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

0,7

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..735

Wikipedia, Partition of a set

Example

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).

Maple

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

Crossrefs

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

Sequence in context: A324163 A127537 A265314 * A307791 A307766 A025443

Adjacent sequences: A364204 A364205 A364206 * A364208 A364209 A364210

Keyword

nonn

Author

Alois P. Heinz, Jul 13 2023

Status

approved