Suppose we have a uniform multinomial distribution over $2^d$ outcomes. Multinomial coefficients give distribution of vector valued variable $v$ where $v$ is the vector of observed counts.
Is there a combinatorial expression for the distribution of $D v$ where $D$ is a matrix obtained by $d$-fold Kronecker product of $A$, a $(-1,0,1)$ valued 2-by-2 matrix?
Three particular choices of $A$ I'm looking at are
$$A_1=\left(\begin{matrix}1&1\\\\1&-1\end{matrix}\right)$$ $$A_2=\left(\begin{matrix}1&1\\\\0&1\end{matrix}\right)$$ $$A_3=\left(\begin{matrix}1&0\\\\-1&1\end{matrix}\right)$$
Motivation: distribution of $Dv$ corresponds to distribution of feature vectors for three commonly feature bases for exponential families (Walsh, Amari's $\eta$ and Amari's $\theta$ bases respectively)
d = 3;
walsh = KroneckerProduct @@ Table[{{1, 1}, {1, -1}}, {d}];
amariEta = KroneckerProduct @@ Table[{{1, 1}, {0, 1}}, {d}];
amariTheta = KroneckerProduct @@ Table[{{1, 0}, {-1, 1}}, {d}];
Edit: all 3 matrices are invertible, so we just need to multiply by appropriate inverse to get back vector of counts and corresponding multinomial coefficient. Inverse of $D$ is just the Kronecker product of inverses of corresponding base matrices
$$A_1^{-1}=\left(\begin{matrix}1/2&1/2\\\\1/2&-1/2\end{matrix}\right)$$ $$A_2^{-1}=\left(\begin{matrix}1&-1\\\\0&1\end{matrix}\right)$$ $$A_3^{-1}=\left(\begin{matrix}1&0\\\\1&1\end{matrix}\right)$$
