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

0,8

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.

Links

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

Example

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].

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 *)

Crossrefs

Cf. A000009 (records), A219339.

Sequence in context: A137350 A334607 A166240 * A114087 A215521 A008284

Adjacent sequences: A219344 A219345 A219346 * A219348 A219349 A219350

Keyword

nonn,look

Author

Alois P. Heinz, Nov 18 2012

Status

approved