A353052 Number of inequivalent {-1,1} matrices of order n, up to permutation of rows and/or columns, multiplication of rows and/or columns by -1, and transposition.

1, 2, 3, 10, 30, 242, 4386

Offset: 1

Author

Nathaniel Johnston, Apr 20 2022

Keywords

• nonn
• hard
• more

Comments

The equivalence operations described in the title are commonly used when discussing Hadamard matrices, for example (see A096201). They are natural when considering norms of these matrices or properties that can be inferred from their singular values, since they do not change singular values. See A352099 for the version of this sequence that does not consider transposition as part of the equivalence relation.

Since the row and column multiplication operations can be used to force the first row and column to consist only of ones, 2^[(n-1)^2] is an upper bound on this sequence. A lower bound is 2^[n*(n-2)] / (n!)^2.

Examples

			When n = 3, there are 3 inequivalent matrices, so a(3) = 3:
  1 1 1       1  1  1       1  1  1
  1 1 1       1  1 -1       1 -1 -1
  1 1 1  and  1 -1 -1  and  1 -1 -1
All other 3-by-3 matrices with entries in {-1,1} can be converted into one of these three matrices by permutating rows and/or columns, multiplying some rows and/or columns by -1, and potentially transposing the matrix.
		

Links

• John Holbrook, Nathaniel Johnston, and Jean-Pierre Schoch, Real Schur norms and Hadamard matrices, arXiv:2206.02863 [math.CO], 2022.
• Nathaniel Johnston, All inequivalent matrices of size 6-by-6 or less

Crossrefs

Cf. A111368, A352099.

Extensions

a(7) from Nathaniel Johnston, May 05 2022