A332722 Index position of [2n-1, 2n-3, ..., 3, 1] within the list of partitions of n^2 in canonical ordering.

1, 1, 2, 9, 74, 711, 7312, 77793, 848557, 9426039, 106218592, 1210785512, 13933358426, 161624712815, 1887635428421, 22176331059637, 261881397819259, 3106736469937751, 37006306302036790, 442425926101676831, 5306994321265281854, 63851605555921588684, 770371217568310624912

Offset

0,3

Comments

The canonical ordering of partitions is described in A080577.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..130

Wikipedia, Integer Partition

Example

a(3) = 9, because 531 has position 9 within the list of partitions of 3*3 in canonical ordering: 9, 81, 72, 711, 63, 621, 6111, 54, 531, ... .

Maple

b:= proc(n, i) option remember;
     `if`(n=0, 1, b(n-i, i-2)+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(n^2$2)-b(n^2, 2*n-1)+1:
seq(a(n), n=0..23);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, b[n - i, i - 2] + 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[n^2, n^2] - b[n^2, 2n - 1] + 1;
a /@ Range[0, 23] (* Jean-François Alcover, May 10 2020, after Maple *)

Crossrefs

Cf. A000041, A000290, A005408, A080577, A238639, A238640, A322761.

Sequence in context: A346648 A336955 A084873 * A103996 A015473 A029849

Adjacent sequences: A332719 A332720 A332721 * A332723 A332724 A332725

Keyword

nonn

Author

Alois P. Heinz, Feb 20 2020

Status

approved