A271425 Number of set partitions of [2n] with maximal block length multiplicity equal to n.

1, 1, 9, 35, 385, 3717, 48279, 691119, 11229075, 200982925, 3928974907, 83060120871, 1885501840677, 45694145548625, 1176704027583075, 32077561625780175, 922854842240358825, 27951355368760441365, 889580295850449177975, 29707539555680924142975

Offset

0,3

Comments

In each set partition of [2n] counted by a(n) at least one block length occurs exactly n times, and all block lengths occur at most n times.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Wikipedia, Partition of a set

Formula

a(n) = A271423(2n,n).

a(n) = A372762(2n,n). - Alois P. Heinz, May 12 2024

Example

a(1) = 1: 12.
a(2) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
a(3) = 35: 123|4|5|6, 124|3|5|6, 12|34|56, 125|3|4|6, 12|35|46, 12|36|45, 126|3|4|5, 134|2|5|6, 13|24|56, 135|2|4|6, 13|25|46, 13|26|45, 136|2|4|5, 14|23|56, 1|234|5|6, 15|23|46, 1|235|4|6, 16|23|45, 1|236|4|5, 145|2|3|6, 14|25|36, 14|26|35, 146|2|3|5, 15|24|36, 1|245|3|6, 16|24|35, 1|246|3|5, 15|26|34, 16|25|34, 1|2|345|6, 1|2|346|5, 156|2|3|4, 1|256|3|4, 1|2|356|4, 1|2|3|456.

Maple

with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
        *b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(2*n$2, n)-b(2*n$2, n-1)):
seq(a(n), n=0..20);

Mathematica

multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]*b[n - i*j, i-1, k]/j!, {j, 0, Min[k, n/i]}]]]; a[n_] := If[n==0, 1, b[2n, 2n, n] - b[2n, 2n, n-1]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 17 2017, translated from Maple *)

Crossrefs

Cf. A271423, A372762.

Sequence in context: A322239 A232282 A364573 * A267081 A020297 A330907

Adjacent sequences: A271422 A271423 A271424 * A271426 A271427 A271428

Keyword

nonn

Author

Alois P. Heinz, Apr 07 2016

Status

approved