Given a Hadamard matrix of size $n$, I want to know what is the minimum number of unique sign patterns in any $\frac n2$ columns (or rows). I count a sign pattern and its negation to be the same.
My guess is exactly $\frac n2$. It is true for the Sylvester construction. Does it remain true in general when their columns and rows are permuted? If yes, any clue on how to prove it formally will be helpful.
On
I recognize this is late, but I have an answer for the minimum number of sign patterns.
Suppose you have a matrix $A\in M_M(\mathbb{R})$. To find the number of sign pattern combinations in the first $\frac{n}{2}$ rows, we first count the number of all possible unique sign patterns with length $n$. We can think of this as a 0-1 list of length $n$, so all possible lists is $2^n$. However, since a sign pattern and its negation are equal, and each sign pattern only has one negation, the total number of unique lists is $2^{n-1}$.
Now, since we are looking at the first $\frac{n}{2}$ rows, we merely choose $\frac{n}{2}$ non-equal patterns of our $2^{n-1}$ list. Hence, the minimum number of sign patterns is exactly $2^{n-1} \choose \frac{n}{2}$.
I'm still not sure I understand the question, but consider the Hadamard matrix $$\matrix{+&+&+&+&+&+&+&+\cr-&+&+&+&-&+&-&-\cr-&-&+&+&+&-&+&-\cr-&-&-&+&+&+&-&+\cr-&+&-&-&+&+&+&-\cr-&-&+&-&-&+&+&+\cr-&+&-&+&-&-&+&+\cr-&+&+&-&+&-&-&+\cr}$$ The first 4 columns have the 8 different sign patterns ++++, -+++, --++, ---+, -+--, --+-, -+-+, and -++-.
If that's not what you mean, please post an example to illustrate your meaning.