Math people:
I would like to verify a conjecture: letting $n \geq 2$, and letting $\mu_1, \mu_2, \ldots, \mu_n, \gamma_1, \gamma_2, \ldots, \gamma_n$ be distinct positive reals with $\gamma_1 < \gamma_2 < \cdots < \gamma_n$, $\vec{\mu} = [\mu_1, \ldots, \mu_n]$ and $\vec{\gamma} = [\gamma_1, \ldots, \gamma_n]$, my conjecture is
$$ f(\vec{\mu},\vec{\gamma})\equiv \sum_{\mathbf{v} \in \{-1,1\}^n} \prod_{i=2}^n \prod_{j=1}^{i-1} (v_i \gamma_i -v_j \gamma_j) \cdot \prod_{i=1}^n \prod_{j=1}^n (\mu_i + v_j \gamma_j) \cdot 2^{\mathbf{v}\cdot \vec{\gamma}} >0. $$
This problem is related to the one I posed at looking for proof or partial proof of determinant conjecture , but I have transformed it, and perhaps someone new will be able to shed some light on it, now that it is in a different form.
The case $n=2$ has been solved, as has the case where all the $\mu$'s are greater than all the $\gamma$'s. I have run a lot of experiments testing the conjecture in Maple, using fractions to avoid roundoff error, and the conjecture has held every single time. I have not tested it, but I suspect the conjecture is also true if you allow the $\mu$'s to be zero or coincide with each other or the $\gamma$'s.
I have run many experiments generating $\vec{\mu}$ and $\vec{\gamma}$ randomly, and looking at $f(s\vec{\mu},\vec{\gamma})$, which is a polynomial of degree $n^2$ in $s$. The coefficients of all the powers of $s$ have always been positive. Thanks to an answer to my previous question, I am 99% sure that the coefficient of $s^{n^2}$ is positive. Unfortunately, that still leaves $n^2 -1$ coefficients. If there is some combinatorial way to show the coefficients of all the powers of $s$ in $f(s\vec{\mu},\vec{\gamma})$ are all positive, that would solve the problem.
This does not look like an easy problem, and I am not taking it for granted that someone will have a complete answer.
For a vector $\mathbf{w}$ of length $n$, does the quantity $\prod_{i=2}^n \prod_{j=1}^{i-1} (w_i - w_j)$ have a name? What is known about it?
