A323347 Number of integer partitions of n whose parts can be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.

1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 6, 2, 11, 2, 7, 7, 10, 2, 18, 2, 17, 13, 9, 2, 50, 3, 10, 24, 34, 2, 85, 2, 51, 46, 12, 9, 261, 2, 13, 80, 257, 2, 258, 2, 323, 431, 15, 2, 1533, 3, 227, 206, 1165, 2, 971, 483, 2409, 309, 18, 2

Offset

0,3

Comments

Rectangles must be of size m X k where m, k are divisors of n and mk <= n. This implies that a(p) = 2 for p prime, since the only allowable rectangles must be of size 1 X 1 corresponding to the partition (p), or 1 X p or p X 1 corresponding to the partition (1,1,...,1). Similarly, a(p^2) = 3 since the allowable rectangles must be of sizes 1 X 1 (partition (p^2)), 1 X p or p X 1 (partition (p,p,...,p)), 1 X p^2, p^2 X 1 and p X p (partition (1,1,...,1)). - Chai Wah Wu, Jan 14 2019

Links

Table of n, a(n) for n=0..59.

Formula

a(p) = 2 and a(p^2) = 3 for p prime (see comment). - Chai Wah Wu, Jan 14 2019

Example

The a(8) = 5 integer partitions are (8), (44), (2222), (3311), (11111111).
The a(12) = 11 integer partitions (C = 12):
  (C)
  (66)
  (444)
  (3333)
  (4422)
  (5511)
  (222222)
  (332211)
  (22221111)
  (222111111)
  (111111111111)
For example, the arrangements of (222111111) are:
  [1 1 2] [1 1 2] [1 2 1] [1 2 1] [2 1 1] [2 1 1]
  [1 2 1] [2 1 1] [1 1 2] [2 1 1] [1 1 2] [1 2 1]
  [2 1 1] [1 2 1] [2 1 1] [1 1 2] [1 2 1] [1 1 2]

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS, facs[n], {2}]), SameQ@@Length/@#&];
Table[Length[Select[IntegerPartitions[n], !Select[ptnmats[Times@@Prime/@#], And[SameQ@@Total/@#, SameQ@@Total/@Transpose[#]]&]=={}&]], {n, 10}]

Crossrefs

A000041(n) = A323348(n) + a(n).

Cf. A006052, A007016, A120733, A319056, A319066, A321719.

Cf. A323300, A323302, A323304, A323306, A323349.

Sequence in context: A272209 A326842 A326843 * A322900 A238791 A007012

Adjacent sequences: A323344 A323345 A323346 * A323348 A323349 A323350

Keyword

nonn,more

Author

Gus Wiseman, Jan 13 2019

Extensions

a(17)-a(53) from Chai Wah Wu, Jan 15 2019

a(54)-a(59) from Chai Wah Wu, Jan 16 2019

Status

approved