OFFSET
1,2
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.
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
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Nathaniel Johnston, Apr 20 2022
EXTENSIONS
a(7) from Nathaniel Johnston, May 05 2022
STATUS
approved