A271739 Number of set partitions of [n] with maximal block length multiplicity equal to ten.

1, 0, 66, 286, 4004, 33033, 328328, 3150576, 31286970, 316394650, 3928974907, 48404715723, 526502083107, 6475762500130, 88834932638892, 1136875206056150, 14448572171583550, 197345257083676845, 2738327374576989195, 37603158111513714720, 528367079280330690400

Offset: 10

Author

Alois P. Heinz, Apr 13 2016

Keywords

• nonn

Comments

At least one block length occurs exactly 10 times, and all block lengths occur at most 10 times.

Links

• Alois P. Heinz, Table of n, a(n) for n = 10..584
• Wikipedia, Partition of a set

Crossrefs

Column k=10 of A271423.

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-> b(n$2, 10)-b(n$2, 9):
seq(a(n), n=10..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, 0, Min[k, n/i] }]]];
a[n_] := b[n, n, 10] - b[n, n, 9];
Table[a[n], {n, 10, 30}] (* Jean-François Alcover, May 08 2018, after Alois P. Heinz *)