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

0,4

Comments

At least one block length occurs exactly once.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..576

Wikipedia, Partition of a set

Formula

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

Example

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.

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 *)

Crossrefs

Column k=1 of A271424.

Sequence in context: A149314 A149315 A149316 * A318120 A355337 A218957

Adjacent sequences: A271423 A271424 A271425 * A271427 A271428 A271429

Keyword

nonn

Author

Alois P. Heinz, Apr 07 2016

Status

approved