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

0, 0, 0, 1, 4, 12, 27, 57, 107, 192, 327, 538, 855, 1329, 2018, 3003, 4402, 6349, 9045, 12720, 17713, 24395, 33335, 45118, 60655, 80888, 107242, 141177, 184905, 240679, 311850, 401860, 515725, 658630, 838006, 1061561, 1340193, 1685271, 2112576, 2638727

Offset

0,5

Comments

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.

References

Sriram Pemmaraju and Steven Skiena, Computational Discrete Mathematics, Cambridge, 2003, page 145.

Links

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

Formula

a(n) = Sum_{k>0} k*A268192(n,k). - Alois P. Heinz, Feb 12 2016

Example

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.

Maple

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    elif i<1 then [0, 0]
    else 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]]
      fi
    end:
a:= n-> add(add(i, i=b(n-j, min(j, n-j)))*j, j=1..n) -n*b(n, n)[1]:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 22 2011, Apr 11 2012

Mathematica

f[list_]:= Total[Select[Reverse[Table[Max[list]-list[[i]], {i, 1, Length[list]}]], #>0&]];
Table[Total[Map[f, IntegerPartitions[n]]], {n, 0, 30}]
(* second program: *)
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]]}]];
a[n_] := Sum[Sum[i, {i, b[n-j, Min[j, n-j]]}]*j, {j, 1, n}]-n*b[n, n][[1]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

Crossrefs

Cf. A066186, A182094, A268192.

Sequence in context: A062479 A327032 A007009 * A104384 A013697 A306055

Adjacent sequences: A188811 A188812 A188813 * A188815 A188816 A188817

Keyword

nonn

Author

Geoffrey Critzer, Apr 22 2011

Status

approved