A271371 Total number of inversions in all partitions of n into distinct parts.

0, 0, 0, 1, 1, 2, 5, 6, 9, 13, 22, 26, 38, 48, 66, 89, 113, 142, 185, 230, 289, 368, 449, 554, 679, 831, 1003, 1224, 1474, 1767, 2117, 2528, 2996, 3568, 4206, 4967, 5855, 6862, 8027, 9391, 10943, 12724, 14785, 17124, 19807, 22898, 26376, 30345, 34893, 40013

Offset

0,6

Links

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..8950 (terms 0..5000 from Alois P. Heinz)

Wikipedia, Inversion (discrete mathematics)

Formula

a(n) = Sum_{k>=1} A161680(k) * A008289(n,k).

Example

a(3) = 1: 21.
a(4) = 1: 31.
a(5) = 2: 32, 41.
a(6) = 5: 42, 51, 321 (three inversions).

Maple

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, [1, 0], b(n, i-1, t)+`if`(i>n, 0,
      (p-> p+[0, p[1]*t])(b(n-i, i-1, t+1)))))
    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 > i*(i+1)/2, 0, If[n == 0, {1, 0}, b[n, i-1, t] + If[i>n, 0, Function[p, p+{0, p[[1]]*t}][b[n-i, i-1, t+1]]]]]; a[n_] := b[n, n, 0][[2]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 05 2017, translated from Maple *)

Crossrefs

Cf. A000009, A008289, A161680, A271370, A271372.

Sequence in context: A190678 A255737 A366741 * A193978 A224486 A163782

Adjacent sequences: A271368 A271369 A271370 * A271372 A271373 A271374

Keyword

nonn

Author

Alois P. Heinz, Apr 05 2016

Status

approved