A332719 Index position of {n}^3 within the list of partitions of 3n in canonical ordering.

1, 3, 8, 19, 44, 93, 187, 357, 657, 1166, 2015, 3393, 5594, 9044, 14378, 22501, 34734, 52931, 79735, 118823, 175337, 256347, 371606, 534377, 762721, 1080979, 1521925, 2129330, 2961580, 4096006, 5634855, 7712558, 10505457, 14243772, 19227383, 25845241, 34600673

Offset: 0

Author

Alois P. Heinz, Feb 20 2020

Keywords

• nonn

Comments

The canonical ordering of partitions is described in A080577.

Examples

			a(2) = 8, because 222 has position 8 within the list of partitions of 6 in canonical ordering: 6, 51, 42, 411, 33, 321, 3111, 222, ... .
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4000
• Wikipedia, Integer Partition

Crossrefs

Cf. A000041, A080577, A322761, A330661, A332706, A332720.

Programs

Maple

b:= proc(n, i) option remember;
     `if`(n=0, 1, b(n-i, i)+g(n, i-1))
    end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
     `if`(i<1, 0, g(n-i, min(n-i, i))+g(n, i-1)))
    end:
a:= n-> g(3*n$2)-b(3*n, n)+1:
seq(a(n), n=0..37);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, b[n - i, i] + g[n, i - 1]];
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, If[i < 1, 0, g[n - i, Min[n - i, i]] + g[n, i - 1]]];
a[n_] := g[3n, 3n] - b[3n, n] + 1;
a /@ Range[0, 37] (* Jean-François Alcover, Jan 06 2021, after Alois P. Heinz *)

Formula

a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*Pi*sqrt(n)). - Vaclav Kotesovec, Feb 28 2020