A271426 Number of set partitions of [n] with minimal block length multiplicity equal to one.

0, 1, 1, 4, 11, 51, 132, 771, 3089, 18388, 96423, 627529, 3349018, 24510305, 155908651, 1171494200, 8647906143, 71603237483, 572103586280, 5172888505403, 43344865682187, 416735802793600, 3830340992280773, 38239507035358011, 374336654847685014

Offset: 0

Author

Alois P. Heinz, Apr 07 2016

Keywords

• nonn

Comments

At least one block length occurs exactly once.

Examples

			a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576
• Wikipedia, Partition of a set

Crossrefs

Column k=1 of A271424.

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$j)
        *b(n-i*j, i-1, k)/j!, j={0,$k..n/i})))
    end:
a:= n-> b(n$2, 1)-b(n$2, 2):
seq(a(n), n=0..30);

Mathematica

multinomial[n_, k_List] := 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}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, Join[{0}, Range[k, n/i]]}]]];
a[n_] := b[n, n, 1] - b[n, n, 2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 07 2018, after Alois P. Heinz *)

Formula

a(n) = A271424(n,1).