A271738 Number of set partitions of [n] with maximal block length multiplicity equal to nine.

1, 0, 55, 220, 2860, 22022, 205205, 1853280, 17381650, 200982925, 2291851991, 23049864630, 262234646310, 3319690300850, 39333605649855, 464026283957060, 5880153732068000, 75836425964702975, 973764622911909400, 12796285021434965050, 173456578124336807300

Offset: 9

Author

Alois P. Heinz, Apr 13 2016

Keywords

• nonn

Comments

At least one block length occurs exactly 9 times, and all block lengths occur at most 9 times.

Links

• Alois P. Heinz, Table of n, a(n) for n = 9..586
• Wikipedia, Partition of a set

Crossrefs

Column k=9 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, 9)-b(n$2, 8):
seq(a(n), n=9..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, 9] - b[n, n, 8];
Table[a[n], {n, 9, 30}] (* Jean-François Alcover, May 08 2018, after Alois P. Heinz *)