A102462 Max{ k!/(a(1)!*a(2)!*..*a(n)!) : a(1) + 2*a(2) + 3*a(3) + ... + n*a(n) = n, a(1) + a(2) + ... + a(n) = k }.

1, 1, 1, 2, 3, 4, 6, 12, 20, 30, 60, 105, 168, 280, 504, 840, 1512, 2520, 5040, 9240, 15840, 27720, 55440, 102960, 180180, 360360, 675675, 1201200, 2162160, 4084080, 7351344, 12697776, 24504480, 46558512, 84651840, 155195040, 296281440, 543182640, 961015440

Offset: 0

Author

Vladeta Jovovic, Feb 23 2005

Keywords

• nonn

Comments

a(n) is the greatest number in row n of A048996 and in row n of A072811. Thus a(n) is the greatest number of compositions (permutations) obtainable from some partition of n. Example: a(7)=12 is the greatest number of compositions from some partition of 7, specifically, the partition {3,2,1,1}. - Clark Kimberling, Dec 24 2006

The partition(s) giving this optimum is always one where #{parts equal to i} >= #{parts equal to j} if i <= j. These partitions are counted in A007294. - Franklin T. Adams-Watters, Apr 08 2008

The number of partition(s) giving this optimum is given by A198254. - Olivier Gérard, Nov 17 2011

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A048992, A059171, A072811, A102356.

Programs

Maple

b:= proc(n,i,p) option remember; `if`(n=0 or i=1, (p+n)!/n!,
       max(seq(b(n-i*j, i-1, p+j)/j!, j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2015

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, (p + n)!/n!, Max[Table[ b[n-i*j, i-1, p+j]/j!, {j, 0, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 19 2015, after Alois P. Heinz *)