A323347 - OEIS

%I #20 Jan 16 2019 19:44:50

%S 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,

%T 85,2,51,46,12,9,261,2,13,80,257,2,258,2,323,431,15,2,1533,3,227,206,

%U 1165,2,971,483,2409,309,18,2

%N 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.

%C 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

%F a(p) = 2 and a(p^2) = 3 for p prime (see comment). - _Chai Wah Wu_, Jan 14 2019

%e The a(8) = 5 integer partitions are (8), (44), (2222), (3311), (11111111).

%e The a(12) = 11 integer partitions (C = 12):

%e   (C)

%e   (66)

%e   (444)

%e   (3333)

%e   (4422)

%e   (5511)

%e   (222222)

%e   (332211)

%e   (22221111)

%e   (222111111)

%e   (111111111111)

%e For example, the arrangements of (222111111) are:

%e   [1 1 2] [1 1 2] [1 2 1] [1 2 1] [2 1 1] [2 1 1]

%e   [1 2 1] [2 1 1] [1 1 2] [2 1 1] [1 1 2] [1 2 1]

%e   [2 1 1] [1 2 1] [2 1 1] [1 1 2] [1 2 1] [1 1 2]

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];

%t Table[Length[Select[IntegerPartitions[n],!Select[ptnmats[Times@@Prime/@#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]],{n,10}]

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

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

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

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Jan 13 2019

%E a(17)-a(53) from _Chai Wah Wu_, Jan 15 2019

%E a(54)-a(59) from _Chai Wah Wu_, Jan 16 2019