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A188814 Sum of the "complements" of the integer partitions of n.

%I A188814 #37 Aug 30 2016 10:19:24
%S A188814 0,0,0,1,4,12,27,57,107,192,327,538,855,1329,2018,3003,4402,6349,9045,
%T A188814 12720,17713,24395,33335,45118,60655,80888,107242,141177,184905,
%U A188814 240679,311850,401860,515725,658630,838006,1061561,1340193,1685271,2112576,2638727
%N A188814 Sum of the "complements" of the integer partitions of n.
%C A188814 Consider the m x k rectangle corresponding to an integer partition p of n, where m is the greatest part of p and k is the number of parts of p.  Fit the Ferrers diagram of p inside its corresponding rectangle.  a(n) is the number of empty spaces in all such rectangles over all partitions of n.
%D A188814 Sriram Pemmaraju and Steven Skiena, Computational Discrete Mathematics, Cambridge, 2003, page 145.
%H A188814 Alois P. Heinz, <a href="/A188814/b188814.txt">Table of n, a(n) for n = 0..1000</a>
%F A188814 a(n) = Sum_{k>0} k*A268192(n,k). - _Alois P. Heinz_, Feb 12 2016
%e A188814 a(4) = 4 because the partitions 4, 2+2, 1+1+1+1 have no empty spaces while the partitions 3+1 and 2+1+1 each have two.
%p A188814 b:= proc(n, i) option remember; local f, g;
%p A188814       if n=0 or i=1 then [1, n]
%p A188814     elif i<1 then [0, 0]
%p A188814     else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
%p A188814          [f[1]+g[1], f[2]+g[2]+g[1]]
%p A188814       fi
%p A188814     end:
%p A188814 a:= n-> add(add(i, i=b(n-j, min(j, n-j)))*j, j=1..n) -n*b(n, n)[1]:
%p A188814 seq(a(n), n=0..40);  # _Alois P. Heinz_, Apr 22 2011, Apr 11 2012
%t A188814 f[list_]:= Total[Select[Reverse[Table[Max[list]-list[[i]],{i,1,Length[list]}]],#>0&]];
%t A188814 Table[Total[Map[f, IntegerPartitions[n]]],{n,0,30}]
%t A188814 (* second program: *)
%t A188814 b[n_, i_] := b[n, i] = Module[{f, g}, If [n==0 || i==1, {1, n}, If[i<1, {0, 0}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]}]];
%t A188814 a[n_] := Sum[Sum[i, {i, b[n-j, Min[j, n-j]]}]*j, {j, 1, n}]-n*b[n,n][[1]];
%t A188814 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Aug 30 2016, after _Alois P. Heinz_ *)
%Y A188814 Cf. A066186, A182094, A268192.
%K A188814 nonn
%O A188814 0,5
%A A188814 _Geoffrey Critzer_, Apr 22 2011