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%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
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