A321724 - OEIS

%I #12 Apr 11 2020 22:42:18

%S 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 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 106,264,42,1,1,30,383,1,1,6,56,1,1,1,1,7,45,321,2804,1731,77,1

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

%C 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 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 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 Andrew Howroyd, <a href="/A321724/b321724.txt">Table of n, a(n) for n = 1..207</a> (rows 1..50)

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Magic_square">Magic square</a>

%F T(n,d) = A333733(d, n/d). - _Andrew Howroyd_, Apr 11 2020

%e Triangle begins:

%e   1

%e   1 1

%e   1 1

%e   1 2 1

%e   1 1

%e   1 2 3 1

%e   1 1

%e   1 3 5 1

%e   1 5 1

%e   1 3 7 1

%e Inequivalent representatives of the a(10,5) = 7 semi-magic squares (zeros not shown):

%e   [2    ] [2    ] [2    ] [2    ] [2    ] [11   ] [11   ]

%e   [ 2   ] [ 2   ] [ 2   ] [ 11  ] [ 11  ] [11   ] [1 1  ]

%e   [  2  ] [  2  ] [  11 ] [ 11  ] [ 1 1 ] [  11 ] [ 1 1 ]

%e   [   2 ] [   11] [  1 1] [   11] [  1 1] [  1 1] [  1 1]

%e   [    2] [   11] [   11] [   11] [   11] [   11] [   11]

%Y Row sums are A321721.

%Y Cf. A006052, A007016, A007716, A057150, A120732, A271103, A319056, A319616.

%Y Cf. A321718, A321719, A321722, A333733.

%K nonn,tabf

%O 1,7

%A _Gus Wiseman_, Nov 18 2018

%E a(28)-a(39) from _Chai Wah Wu_, Jan 16 2019

%E Terms a(40) and beyond from _Andrew Howroyd_, Apr 11 2020