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A321724 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic non-normal semi-magic square multiset partitions of weight n and length d = A027750(n, k).

%I A321724 #21 Mar 05 2025 22:04:46
%S A321724 1,1,1,1,1,1,2,1,1,1,1,2,3,1,1,1,1,3,5,1,1,5,1,1,3,7,1,1,1,1,4,9,12,
%T A321724 11,1,1,1,1,4,15,1,1,13,31,1,1,5,43,22,1,1,1,1,5,22,103,30,1,1,1,1,6,
%U A321724 106,264,42,1,1,30,383,1,1,6,56,1,1,1,1,7,45,321,2804,1731,77,1
%N A321724 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic non-normal semi-magic square multiset partitions of weight n and length d = A027750(n, k).
%C A321724 Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with row sums and column sums all equal to d.
%C A321724 A non-normal semi-magic square multiset partition of weight n is a multiset partition of weight n whose part sizes and vertex degrees are all equal to d, for some d|n.
%C A321724 The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%H A321724 Andrew Howroyd, <a href="/A321724/b321724.txt">Table of n, a(n) for n = 1..207</a> (rows 1..50)
%H A321724 Wikipedia, <a href="https://en.wikipedia.org/wiki/Magic_square">Magic square</a>
%F A321724 T(n,k) = A333733(d, n/d), where d = A027750(n, k). - _Andrew Howroyd_, Apr 11 2020
%e A321724 Triangle begins:
%e A321724   1
%e A321724   1 1
%e A321724   1 1
%e A321724   1 2 1
%e A321724   1 1
%e A321724   1 2 3 1
%e A321724   1 1
%e A321724   1 3 5 1
%e A321724   1 5 1
%e A321724   1 3 7 1
%e A321724 Inequivalent representatives of the T(10,3) = 7 semi-magic squares (zeros not shown):
%e A321724   [2    ] [2    ] [2    ] [2    ] [2    ] [11   ] [11   ]
%e A321724   [ 2   ] [ 2   ] [ 2   ] [ 11  ] [ 11  ] [11   ] [1 1  ]
%e A321724   [  2  ] [  2  ] [  11 ] [ 11  ] [ 1 1 ] [  11 ] [ 1 1 ]
%e A321724   [   2 ] [   11] [  1 1] [   11] [  1 1] [  1 1] [  1 1]
%e A321724   [    2] [   11] [   11] [   11] [   11] [   11] [   11]
%Y A321724 Row sums are A321721.
%Y A321724 Cf. A006052, A007016, A007716, A027750, A057150, A120732, A271103, A319056, A319616.
%Y A321724 Cf. A321718, A321719, A321722, A333733.
%K A321724 nonn,tabf
%O A321724 1,7
%A A321724 _Gus Wiseman_, Nov 18 2018
%E A321724 a(28)-a(39) from _Chai Wah Wu_, Jan 16 2019
%E A321724 Terms a(40) and beyond from _Andrew Howroyd_, Apr 11 2020
%E A321724 Edited by _Peter Munn_, Mar 05 2025