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Total number of inversions in all partitions of n.
5

%I #16 Feb 03 2017 13:16:52

%S 0,0,0,1,3,9,18,38,68,120,200,326,508,785,1179,1741,2532,3633,5141,

%T 7199,9972,13680,18618,25116,33642,44738,59139,77653,101444,131751,

%U 170320,219049,280553,357652,454254,574507,724135,909265,1138169,1419737,1765884,2189441

%N Total number of inversions in all partitions of n.

%H Alois P. Heinz, <a href="/A271370/b271370.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Inversion_(discrete_mathematics)">Inversion (discrete mathematics)</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

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

%e a(3) = 1: one inversion in 21.

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

%p b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,

%p add((p-> p+[0, p[1]*j*t])(b(n-i*j, i-1, t+j)), j=0..n/i)))

%p end:

%p a:= n-> b(n$2, 0)[2]:

%p seq(a(n), n=0..60);

%t 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}]]];

%t a[n_] := b[n, n, 0][[2]]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Feb 03 2017, translated from Maple *)

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

%K nonn

%O 0,5

%A _Alois P. Heinz_, Apr 05 2016