A271371 -id:A271371 - cp's OEIS frontend

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

Alois P. Heinz, Apr 05 2016

nonn

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

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

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

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);

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 *)

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

A271372 Total number of inversions in all compositions of n into distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 11, 12, 21, 31, 112, 122, 212, 294, 456, 1147, 1381, 2144, 3059, 4494, 6081, 13597, 15928, 24716, 33728, 49260, 65016, 93229, 169249, 210206, 304979, 417600, 584037, 779731, 1076824, 1409102, 2418068, 2950722, 4213584, 5581351, 7779829

Offset: 0

Alois P. Heinz, Apr 05 2016

nonn

			a(3) = 1: 21.
a(4) = 1: 31.
a(5) = 2: 41, 32.
a(6) = 11: one inversion in each of 51, 132, 42, 213, two inversions in each of 231, 312, three inversions in 321.

Alois P. Heinz, Table of n, a(n) for n = 0..6000
Wikipedia, Inversion (discrete mathematics)

Cf. A001809, A008289, A032020, A189052, A271371.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, t!*t*(t-1)/4, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t!*t*(t - 1)/4, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t + 1]]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 29 2018, from Maple *)

a(n) = Sum_{k>=1} A001809(k) * A008289(n,k).

cp's OEIS Frontend

A271370 Total number of inversions in all partitions of n.

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