A271370 Total number of inversions in all partitions of n.

0, 0, 0, 1, 3, 9, 18, 38, 68, 120, 200, 326, 508, 785, 1179, 1741, 2532, 3633, 5141, 7199, 9972, 13680, 18618, 25116, 33642, 44738, 59139, 77653, 101444, 131751, 170320, 219049, 280553, 357652, 454254, 574507, 724135, 909265, 1138169, 1419737, 1765884, 2189441

Offset: 0

Author

Alois P. Heinz, Apr 05 2016

Keywords

• nonn

Examples

			a(3) = 1: one inversion in 21.
a(4) = 3: one inversion in 31, and two inversions in 211.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Inversion (discrete mathematics)
• Wikipedia, Partition (number theory)

Crossrefs

Cf. A000041, A189052, A264033, A264082, A271371.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+[0, p[1]*j*t])(b(n-i*j, i-1, t+j)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..60);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n==0, {1, 0}, If[i<1, 0, Sum[Function[p, If[p === 0, 0, p+{0, p[[1]]*j*t}]][b[n-i*j, i-1, t+j]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0][[2]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 03 2017, translated from Maple *)

Formula

a(n) = Sum_{k>0} k * A264033(n,k).