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Number of partitions of n into distinct parts with smallest possible largest part.
2

%I #22 Dec 10 2020 03:27:34

%S 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,

%T 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,

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

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

%C 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.

%H Alois P. Heinz, <a href="/A219347/b219347.txt">Table of n, a(n) for n = 0..10000</a>

%e a(0) = 1: [].

%e a(7) = 2: [4,2,1], [4,3].

%e a(16) = 3: [6,4,3,2,1], [6,5,3,2], [6,5,4,1].

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

%p g:= proc(n, i) option remember; local s; s:=i*(i+1)/2;

%p `if`(n=s, 1, `if`(n>s, 0, g(n, i-1)+ `if`(i>n, 0, g(n-i, i-1))))

%p end:

%p a:= n-> g(n, floor(sqrt(2*n)+1/2)):

%p seq (a(n), n=0..120);

%t 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]]]]];

%t a[n_] := g[n, Floor[Sqrt[2n] + 1/2]];

%t a /@ Range[0, 120] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)

%Y Cf. A000009 (records), A219339.

%K nonn,look

%O 0,8

%A _Alois P. Heinz_, Nov 18 2012