A182099 Total area of the largest inscribed rectangles of all integer partitions of n.

0, 1, 4, 8, 18, 29, 54, 82, 136, 202, 309, 441, 658, 915, 1303, 1790, 2479, 3337, 4541, 6022, 8045, 10554, 13876, 17996, 23409, 30055, 38634, 49208, 62650, 79116, 99898, 125213, 156848, 195339, 242964, 300707, 371770, 457493, 562292, 688451, 841707, 1025484

Offset: 0

Author

Alois P. Heinz, Apr 11 2012

Keywords

• nonn

Comments

a(n) >= A000041(n)*A061017(n) for n>0 because the least largest inscribed rectangle of any integer partition of n is A061017(n) and A000041(n) is the number of partitions of n.

a(n) >= A116503(n), the sum of the areas of the Durfee squares of all partitions of n.

Examples

			a(4) = 18 = 4+3+4+3+4 because the partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4] and the largest inscribed rectangles have areas 4*1, 3*1, 2*2, 1*3, 1*4.
a(5) = 29 = 5+4+4+3+4+4+5 because the partitions of 5 are [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..175

Crossrefs

Cf. A000041, A061017, A115723, A116503.

Programs

Maple

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i=1, `if`(t+n>k, 0, 1), `if`(i<1, 0, b(n, i-1, t, k)
      +add(`if`(t+j>k/i, 0, b(n-i*j, i-1, t+j, k)), j=1..n/i))))
    end:
a:= n-> add(k*(b(n, n, 0, k) -b(n, n, 0, k-1)), k=1..n):
seq(a(n), n=0..50);

Mathematica

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i == 1, If[t + n > k, 0, 1], If[i < 1, 0, b[n, i - 1, t, k] + Sum[If[t + j > k/i, 0, b[n - i j, i - 1, t + j, k]], {j, 1, n/i}]]]];
a[n_] := Sum[k(b[n, n, 0, k] - b[n, n, 0, k - 1]), {k, 1, n}];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 06 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} k * A115723(n,k) for n>0, a(0) = 0.

a(n) = Sum_{k=1..n} k * (A182114(n,k) - A182114(n,k-1)).