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%I #23 Dec 19 2015 14:27:45
%S 1,1,1,2,3,4,6,12,20,30,60,105,168,280,504,840,1512,2520,5040,9240,
%T 15840,27720,55440,102960,180180,360360,675675,1201200,2162160,
%U 4084080,7351344,12697776,24504480,46558512,84651840,155195040,296281440,543182640,961015440
%N 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 }.
%C 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
%C 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
%C The number of partition(s) giving this optimum is given by A198254. - _Olivier Gérard_, Nov 17 2011
%H Alois P. Heinz, <a href="/A102462/b102462.txt">Table of n, a(n) for n = 0..1000</a>
%p b:= proc(n,i,p) option remember; `if`(n=0 or i=1, (p+n)!/n!,
%p max(seq(b(n-i*j, i-1, p+j)/j!, j=0..n/i)))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Apr 15 2015
%t 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_ *)
%Y Cf. A048992, A059171, A072811, A102356.
%K nonn
%O 0,4
%A _Vladeta Jovovic_, Feb 23 2005
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