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

Author

Alois P. Heinz, Apr 07 2016

Keywords

• nonn

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.

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Wikipedia, Partition of a set

Crossrefs

Cf. A271423, A372762.

Programs

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 *)

Formula

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

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