A219347 Number of partitions of n into distinct parts with smallest possible largest part.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 5, 4, 3, 2, 2, 1, 1, 1, 6, 5, 4, 3, 2, 2, 1, 1, 1, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1, 10, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1, 12, 10, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1, 15, 12, 10, 8, 6, 5

Offset: 0

Author

Alois P. Heinz, Nov 18 2012

Keywords

• nonn
• look

Comments

Size of the smallest possible largest part is floor(sqrt(2*n)+1/2) = A002024(n). Records occur at 0, 7, and A000124(k) for k>=5.

Examples

			a(0) = 1: [].
a(7) = 2: [4,2,1], [4,3].
a(16) = 3: [6,4,3,2,1], [6,5,3,2], [6,5,4,1].
a(22) = 4: [7,5,4,3,2,1], [7,6,4,3,2], [7,6,5,3,1], [7,6,5,4].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000009 (records), A219339.

Programs

Maple

g:= proc(n, i) option remember; local s; s:=i*(i+1)/2;
      `if`(n=s, 1, `if`(n>s, 0, g(n, i-1)+ `if`(i>n, 0, g(n-i, i-1))))
    end:
a:= n-> g(n, floor(sqrt(2*n)+1/2)):
seq (a(n), n=0..120);

Mathematica

g[n_, i_] := g[n, i] = Module[{s = i(i+1)/2}, If[n == s, 1, If[n > s, 0, g[n, i - 1] + If[i > n, 0, g[n - i, i - 1]]]]];
a[n_] := g[n, Floor[Sqrt[2n] + 1/2]];
a /@ Range[0, 120] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)