A323348 Number of integer partitions of n whose parts cannot be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.

0, 0, 0, 1, 2, 5, 6, 13, 17, 27, 36, 54, 66, 99, 128, 169, 221, 295, 367, 488, 610, 779, 993, 1253, 1525, 1955, 2426, 2986, 3684, 4563, 5519, 6840, 8298, 10097, 12298, 14874, 17716, 21635, 26002, 31105, 37081, 44581, 52916, 63259, 74852, 88703, 105543, 124752, 145740, 173522, 203999, 239737, 280424, 329929

Offset

0,5

Links

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

Example

The a(8) = 17 integer partitions:
  (53), (62), (71),
  (332), (422), (431), (521), (611),
  (3221), (4211), (5111),
  (22211), (32111), (41111),
  (221111), (311111),
  (2111111).

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) = A323347(n) + a(n).

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

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

Sequence in context: A327472 A283684 A325285 * A181314 A027010 A038191

Adjacent sequences: A323345 A323346 A323347 * A323349 A323350 A323351

Keyword

nonn

Author

Gus Wiseman, Jan 13 2019

Extensions

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

Status

approved