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

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Wikipedia, Inversion (discrete mathematics)

Wikipedia, Partition (number theory)

Formula

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A264365 A291143 A057578 * A015635 A062882 A363534

Adjacent sequences: A271367 A271368 A271369 * A271371 A271372 A271373

Keyword

nonn

Author

Alois P. Heinz, Apr 05 2016

Status

approved