A266871 Number of partitions of n that maximize the product of multiplicities of parts.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1

Offset: 0

Author

Emeric Deutsch and Alois P. Heinz, Jan 04 2016

Keywords

• nonn

Examples

			a(8) = 2: [1,1,1,1,1,1,1,1], [1,1,1,1,2,2] (product of multiplicities = 8).
a(9) = 1: [1,1,1,1,1,2,2] (product = 10).
a(10) = 2: [1,1,1,1,1,1,2,2], [1,1,1,1,2,2,2] (product = 12).
a(11) = 1: [1,1,1,1,1,2,2,2] (product = 15).
a(23) = 3: [1,1,1,1,1,1,1,1,1,2,2,2,2,3,3], [1,1,1,1,1,1,1,1,2,2,2,3,3,3], [1,1,1,1,1,1,2,2,2,2,3,3,3] (product = 72).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A266477, A266480.

Programs

Maple

b:= proc(n, i) option remember; local r,l,j;
      if n=0 or i=1 then [max(1, n),1]
    else r:= b(n, i-1);
         for j to iquo(n, i) do
           l:= (w-> [w[1]*j, w[2]])(b(n-i*j, i-1));
           r:= `if`(l[1]>r[1], l,
               `if`(l[1]=r[1], [0, l[2]], 0)+r)
         od; r
      fi
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..120);

Mathematica

b[n_, i_] := b[n, i] = Module[{r, l, j}, If[n == 0 || i == 1, {Max[1, n], 1}, r = b[n, i - 1]; For[j = 1, j <= Quotient[n, i], j++, l = Function[w, {w[[1]]*j, w[[2]]}][b[n - i*j, i - 1]]; r = If[l[[1]] > r[[1]], l, If[l[[1]] == r[[1]], {0, l[[2]]}, 0] + r]]; r]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Dec 21 2016, translated from Maple *)

Formula

a(n) = A266477(n,A266480(n)).