A323523 -id:A323523 - cp's OEIS frontend

A323522 Number of ways to fill a square matrix with the parts of a strict integer partition of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 25, 49, 73, 121, 145, 217, 265, 361, 433, 553, 649, 817, 937, 1129, 1297, 1537, 1729, 2017, 2257, 2593, 2881, 3265, 3601, 4057, 4441, 4945, 5401, 5977, 6481, 7129, 7705, 8425, 9073, 9865, 373465, 374353, 738025, 1101865, 1828513

Offset: 0

Gus Wiseman, Jan 17 2019

nonn

			The a(10) = 25 matrices:
  [10]
.
  [4 3] [4 3] [4 2] [4 2] [4 1] [4 1] [3 4] [3 4]
  [2 1] [1 2] [3 1] [1 3] [3 2] [2 3] [2 1] [1 2]
.
  [3 2] [3 2] [3 1] [3 1] [2 4] [2 4] [2 3] [2 3]
  [4 1] [1 4] [4 2] [2 4] [3 1] [1 3] [4 1] [1 4]
.
  [2 1] [2 1] [1 4] [1 4] [1 3] [1 3] [1 2] [1 2]
  [4 3] [3 4] [3 2] [2 3] [4 2] [2 4] [4 3] [3 4]

Alois P. Heinz, Table of n, a(n) for n = 0..7000

Cf. A000009, A089299, A103198 (non-strict case), A120732, A323431, A323434, A323519, A323523, A323525, A323529.

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
    end:
a:= n-> (l-> add(l[i^2+1]*(i^2)!, i=0..floor(sqrt(nops(l)-1))))(b(n$2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 17 2019

Table[Sum[(k^2)!*Length[Select[IntegerPartitions[n,{k^2}],UnsameQ@@#&]],{k,n}],{n,20}]
(* Second program: *)
q[n_, k_] := q[n, k] = If[n < k || k < 1, 0,
     If[n == 1, 1, q[n-k, k] + q[n-k, k-1]]];
a[n_] := If[n == 0, 1, Sum[(k^2)! q[n, k^2], {k, 0, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 20 2021 *)

a(n) = Sum_{k >= 0} (k^2)! * Q(n, k^2) where Q = A008289.

A323524 Number of integer partitions of n whose parts can be arranged into a square matrix with equal row and column sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 5, 1, 4, 4, 6, 1, 10, 1, 7, 10, 6, 1, 24, 2, 7, 22, 18, 1, 38, 1, 35, 43, 9, 6, 124, 1, 10, 77, 158, 1, 110, 1, 285, 186, 12, 1, 742, 2, 170, 203, 1110, 1, 285, 480, 2115, 306, 15, 1

Offset: 0

Gus Wiseman, Jan 17 2019

			The a(12) = 5 integer partitions are (12), (5,5,1,1), (4,4,2,2), (3,3,3,3), (2,2,2,1,1,1,1,1,1). For example, such a matrix for (2,2,2,1,1,1,1,1,1) is:
  [1 1 2]
  [2 1 1]
  [1 2 1]

Cf. A006052, A007016, A103198, A120732, A321719, A321722, A321725, A323306, A323347, A323349, A323523.

a(p) = 1 and a(p^2) = 2 for p prime (see comment in A323349). - Chai Wah Wu, Jan 20 2019

a(16)-a(59) from Chai Wah Wu, Jan 20 2019

cp's OEIS Frontend

A323522 Number of ways to fill a square matrix with the parts of a strict integer partition of n.

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