A271737 Number of set partitions of [n] with maximal block length multiplicity equal to eight.

1, 0, 45, 165, 1980, 14157, 123123, 1042470, 11229075, 117721175, 1085614101, 11354532696, 132028149240, 1440550986525, 15693895739115, 183700174158435, 2200557929261230, 26295830857171150, 323510486572841425, 4085513198322259275, 52716487743732737925

Offset: 8

Author

Alois P. Heinz, Apr 13 2016

Keywords

• nonn

Comments

At least one block length occurs exactly 8 times, and all block lengths occur at most 8 times.

Links

• Alois P. Heinz, Table of n, a(n) for n = 8..588
• Wikipedia, Partition of a set

Crossrefs

Column k=8 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, 8)-b(n$2, 7):
seq(a(n), n=8..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, 8] - b[n, n, 7];
Table[a[n], {n, 8, 30}] (* Jean-François Alcover, May 08 2018, after Alois P. Heinz *)