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A323349 Number of positive integer matrices with entries summing to n, with equal row-sums and equal column-sums.

%I A323349 #36 Dec 30 2023 21:24:37
%S A323349 1,1,3,3,6,3,11,3,12,6,13,3,52,3,15,30,57,3,156,3,238,129,19,3,2221,6,
%T A323349 21,415,3114,3,14921,3,12853,1044,25,6219,164743,3,27,2220,851476,3,
%U A323349 954088,3,434106,3326714,31,3,24648724,6,22309800,7269,2737618,3,69823653
%N A323349 Number of positive integer matrices with entries summing to n, with equal row-sums and equal column-sums.
%C A323349 Also the number of non-normal semi-magic rectangles summing to n with no zeros.
%C A323349 Matrices must be of size m X k where m, k are divisors of n and mk <= n. This implies that a(p) = 3 for p prime, since the only allowable matrices must be of size 1 X 1, 1 X p or p X 1 with only one way to fill in the entries for each matrix size. Similarly, a(p^2) = 6 with additional allowable matrices of sizes 1 X p^2, p^2 X 1 and p X p, again with only one way to fill in the entries for each size. - _Chai Wah Wu_, Jan 13 2019
%H A323349 Chai Wah Wu, <a href="/A323349/b323349.txt">Table of n, a(n) for n = 0..59</a>
%F A323349 a(p) = 3 and a(p^2) = 6 for p prime (see comment). - _Chai Wah Wu_, Jan 13 2019
%e A323349 The a(6) = 11 matrices:
%e A323349   [6] [3 3] [2 2 2] [1 1 1 1 1 1]
%e A323349 .
%e A323349   [3] [1 2] [2 1] [1 1 1]
%e A323349   [3] [2 1] [1 2] [1 1 1]
%e A323349 .
%e A323349   [2] [1 1]
%e A323349   [2] [1 1]
%e A323349   [2] [1 1]
%e A323349 .
%e A323349   [1]
%e A323349   [1]
%e A323349   [1]
%e A323349   [1]
%e A323349   [1]
%e A323349   [1]
%t A323349 Table[Length[Select[Join@@Table[Partition[cmp,d],{cmp,Join@@Permutations/@IntegerPartitions[n]},{d,Divisors[Length[cmp]]}],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]],{n,10}]
%Y A323349 Cf. A000219, A006052, A120733, A305551, A319056, A321719, A323295, A323300, A323302, A323306, A323347, A323349.
%K A323349 nonn
%O A323349 0,3
%A A323349 _Gus Wiseman_, Jan 13 2019
%E A323349 a(21)-a(31) from _Chai Wah Wu_, Jan 13 2019
%E A323349 a(32)-a(53) from _Chai Wah Wu_, Jan 14 2019
%E A323349 a(54) from _Chai Wah Wu_, Jan 16 2019