A321721 Number of non-isomorphic non-normal semi-magic square multiset partitions of weight n.

1, 1, 2, 2, 4, 2, 7, 2, 10, 7, 12, 2, 38, 2, 21, 46, 72, 2, 162, 2, 420, 415, 64, 2, 4987, 1858, 110, 9336, 45456, 2, 136018, 2, 1014658, 406578, 308, 3996977, 34937078, 2, 502, 28010167, 1530292965, 2, 508164038, 2, 54902992348, 51712929897, 1269, 2, 3217847072904, 8597641914, 9168720349613

Offset

0,3

Comments

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.

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, for some d|n.

Links

Table of n, a(n) for n=0..50.

Wikipedia, Magic square

Formula

a(p) = 2 for p prime corresponding to the 1 X 1 square [p] and the permutation matrices of size p X p with partition (1...10...0). - Chai Wah Wu, Jan 16 2019

a(n) = Sum_{d|n} A333733(d,n/d) for n > 0. - Andrew Howroyd, Apr 11 2020

Example

Non-isomorphic representatives of the a(2) = 2 through a(6) = 7 multiset partitions:
  {{11}}   {{111}}     {{1111}}       {{11111}}         {{111111}}
  {{1}{2}} {{1}{2}{3}} {{11}{22}}     {{1}{2}{3}{4}{5}} {{111}{222}}
                       {{12}{12}}                       {{112}{122}}
                       {{1}{2}{3}{4}}                   {{11}{22}{33}}
                                                        {{11}{23}{23}}
                                                        {{12}{13}{23}}
                                                        {{1}{2}{3}{4}{5}{6}}
Inequivalent representatives of the a(6) = 7 matrices:
  [6]
.
  [3 0] [2 1]
  [0 3] [1 2]
.
  [2 0 0] [2 0 0] [1 1 0]
  [0 2 0] [0 1 1] [1 0 1]
  [0 0 2] [0 1 1] [0 1 1]
.
  [1 0 0 0 0 0]
  [0 1 0 0 0 0]
  [0 0 1 0 0 0]
  [0 0 0 1 0 0]
  [0 0 0 0 1 0]
  [0 0 0 0 0 1]
Inequivalent representatives of the a(9) = 7 matrices:
  [9]
.
  [3 0 0] [3 0 0] [2 1 0] [2 1 0] [1 1 1]
  [0 3 0] [0 2 1] [1 1 1] [1 0 2] [1 1 1]
  [0 0 3] [0 1 2] [0 1 2] [0 2 1] [1 1 1]
.
  [1 0 0 0 0 0 0 0 0]
  [0 1 0 0 0 0 0 0 0]
  [0 0 1 0 0 0 0 0 0]
  [0 0 0 1 0 0 0 0 0]
  [0 0 0 0 1 0 0 0 0]
  [0 0 0 0 0 1 0 0 0]
  [0 0 0 0 0 0 1 0 0]
  [0 0 0 0 0 0 0 1 0]
  [0 0 0 0 0 0 0 0 1]

Crossrefs

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

Cf. A321717, A321718, A321719, A321722, A321724, A333733.

Sequence in context: A348219 A122977 A275870 * A359102 A003980 A286369

Adjacent sequences: A321718 A321719 A321720 * A321722 A321723 A321724

Keyword

nonn

Author

Gus Wiseman, Nov 18 2018

Extensions

a(11)-a(13) from Chai Wah Wu, Jan 16 2019

a(14)-a(15) from Chai Wah Wu, Jan 20 2019

Terms a(16) and beyond from Andrew Howroyd, Apr 11 2020

Status

approved