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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.

%I A323348 #15 Jan 16 2019 19:45:27
%S A323348 0,0,0,1,2,5,6,13,17,27,36,54,66,99,128,169,221,295,367,488,610,779,
%T A323348 993,1253,1525,1955,2426,2986,3684,4563,5519,6840,8298,10097,12298,
%U A323348 14874,17716,21635,26002,31105,37081,44581,52916,63259,74852,88703,105543,124752,145740,173522,203999,239737,280424,329929
%N 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.
%H A323348 Chai Wah Wu, <a href="/A323348/b323348.txt">Table of n, a(n) for n = 0..59</a>
%e A323348 The a(8) = 17 integer partitions:
%e A323348   (53), (62), (71),
%e A323348   (332), (422), (431), (521), (611),
%e A323348   (3221), (4211), (5111),
%e A323348   (22211), (32111), (41111),
%e A323348   (221111), (311111),
%e A323348   (2111111).
%t A323348 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A323348 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A323348 ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
%t A323348 Table[Length[Select[IntegerPartitions[n],Select[ptnmats[Times@@Prime/@#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]],{n,10}]
%Y A323348 A000041(n) = A323347(n) + a(n).
%Y A323348 Cf. A006052, A007016, A120733, A319056, A319066, A321719.
%Y A323348 Cf. A323300, A323302, A323304, A323306, A323349.
%K A323348 nonn
%O A323348 0,5
%A A323348 _Gus Wiseman_, Jan 13 2019
%E A323348 a(17)-a(53) from _Chai Wah Wu_, Jan 15 2019