A271736 Number of set partitions of [n] with maximal block length multiplicity equal to seven.

1, 0, 36, 120, 1320, 8712, 70356, 691119, 6628050, 55398200, 528441056, 5607882072, 55953959256, 559256993400, 6033783063160, 66852986570260, 743874599106485, 8455383000184208, 100088596628849400, 1202568046655647100, 14764362076427728050

Offset: 7

Author

Alois P. Heinz, Apr 13 2016

Keywords

• nonn

Comments

At least one block length occurs exactly 7 times, and all block lengths occur at most 7 times.

Links

• Alois P. Heinz, Table of n, a(n) for n = 7..592
• Wikipedia, Partition of a set

Crossrefs

Column k=7 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, 7)-b(n$2, 6):
seq(a(n), n=7..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, 7] - b[n, n, 6];
Table[a[n], {n, 7, 30}] (* Jean-François Alcover, May 08 2018, after Alois P. Heinz *)