A115723 -id:A115723 - cp's OEIS frontend

A182114 Number of partitions of n with largest inscribed rectangle having area <= k; triangle T(n,k), 0<=n, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 2, 5, 0, 0, 0, 1, 5, 7, 0, 0, 0, 0, 5, 7, 11, 0, 0, 0, 0, 3, 7, 13, 15, 0, 0, 0, 0, 1, 5, 16, 18, 22, 0, 0, 0, 0, 0, 3, 17, 21, 27, 30, 0, 0, 0, 0, 0, 1, 16, 22, 34, 38, 42, 0, 0, 0, 0, 0, 0, 13, 21, 39, 48, 54, 56

Offset: 0

Alois P. Heinz, Apr 12 2012

T(n,k) = A000041(k) for n

Sum_{n>=0} T(n,k) = A115725(k).

			T(5,4) = 5 because there are 5 partitions of 5 with largest inscribed rectangle having area <= 4: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4].
T(9,5) = 3: [1,1,1,2,4], [1,1,1,1,5], [1,1,2,5].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 1, 3;
  0, 0, 0, 2, 5;
  0, 0, 0, 1, 5, 7;
  0, 0, 0, 0, 5, 7, 11;
  0, 0, 0, 0, 3, 7, 13, 15;
  0, 0, 0, 0, 1, 5, 16, 18, 22;
  0, 0, 0, 0, 0, 3, 17, 21, 27, 30;
  0, 0, 0, 0, 0, 1, 16, 22, 34, 38, 42;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Diagonal gives: A000041.
T(n,n-1) = A144300(n) = A000041(n) - A000005(n).
T(n+d,n) for d=2-10 give: A218623, A218624, A218625, A218626, A218627, A218628, A218629, A218630, A218631.
Cf. A115723, A115725.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i=1, `if`(t+n<=k, 1, 0), `if`(i<1, 0, b(n, i-1, t, k)+
       add(`if`(t+j<=k/i, b(n-i*j, i-1, t+j, k), 0), j=1..n/i))))
    end:
T:= (n, k)-> b(n, n, 0, k):
seq(seq(T(n, k), k=0..n), n=0..15);

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i == 1, If[t + n <= k, 1, 0], If[i < 1, 0, b[n, i - 1, t, k] + Sum[If[t + j <= k/i, b[n - i*j, i - 1, t + j, k], 0], {j, 1, n/i}]]]] ; T[n_, k_] := b[n, n, 0, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 15}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

T(n,k) = Sum_{i=1..k} A115723(n,i) for n>0, T(0,0) = 1.

A115724 Number of partitions with maximum rectangle n.

Original entry on oeis.org

1, 1, 3, 5, 16, 16, 76, 53, 218, 303, 750, 412, 3680, 1361, 5015, 9206, 23162, 8290, 66166, 19936, 161656, 192181, 236007, 100730, 1338186, 819694, 1180478, 1924986, 5215844, 1246468, 17370367, 3098322, 24926724, 23473968, 24790503, 41886304, 227243488

Offset: 0

Franklin T. Adams-Watters, Mar 11 2006

nonn

			The 16 partitions with maximum rectangle 4 are [4], [2^2], [1^4], [4,1], [3,2], [2^2,1], [2,1^3], [4,2], [4,1^2], [3,2,1], [3,1^3], [2^2,1^2], [4,2,1], [4,1^3], [3,2,1^2] and [4,2,1^2].

Alois P. Heinz, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Ferrers Diagram.

Cf. A115725, A115723, A115726, A115727.

a(n) = A115725(n) - A115725(n-1).

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

Original entry on oeis.org

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

Alois P. Heinz, Apr 11 2012

nonn

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.

			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].

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

Cf. A000041, A061017, A115723, A116503.

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);

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 *)

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)).

Showing 1-3 of 3 results.

cp's OEIS Frontend

A182114 Number of partitions of n with largest inscribed rectangle having area <= k; triangle T(n,k), 0<=n, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A115724 Number of partitions with maximum rectangle n.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula