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).

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

Offset: 1

Author

Gus Wiseman, Nov 18 2018

Keywords

• nonn
• tabf

Comments

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.

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.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Triangle begins:
  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
Inequivalent representatives of the T(10,3) = 7 semi-magic squares (zeros not shown):
  [2    ] [2    ] [2    ] [2    ] [2    ] [11   ] [11   ]
  [ 2   ] [ 2   ] [ 2   ] [ 11  ] [ 11  ] [11   ] [1 1  ]
  [  2  ] [  2  ] [  11 ] [ 11  ] [ 1 1 ] [  11 ] [ 1 1 ]
  [   2 ] [   11] [  1 1] [   11] [  1 1] [  1 1] [  1 1]
  [    2] [   11] [   11] [   11] [   11] [   11] [   11]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..207 (rows 1..50)
• Wikipedia, Magic square

Crossrefs

Row sums are A321721.

Cf. A006052, A007016, A007716, A027750, A057150, A120732, A271103, A319056, A319616.

Cf. A321718, A321719, A321722, A333733.

Formula

T(n,k) = A333733(d, n/d), where d = A027750(n, k). - Andrew Howroyd, Apr 11 2020

Extensions

a(28)-a(39) from Chai Wah Wu, Jan 16 2019

Terms a(40) and beyond from Andrew Howroyd, Apr 11 2020

Edited by Peter Munn, Mar 05 2025