Totally non-negativity of a family of change-of-basis matrices

Let $\mathbf{a}=(a_1,a_2,\dots ,a_n)$ and $\mathbf{e}=(e_1,e_2,\dots ,e_n)$ be real sequences. Denote by $M_{\mathbf{e}\to \mathbf{a}}$ the $(n+1)\times (n+1)$ matrix whose $(m,k)$ entry ($m,k\in \{0,\dots ,n\}$) is the coefficient of the polynomial $(x-a_1)\cdots (x-a_k)$ in the expansion of $(x-e_1)\cdots (x-e_m)$ as a linear combination of the polynomials $1,x-a_1,\dots ,(x-a_1)\cdots (x-a_m)$. By appropriate choice of $\mathbf{a}$ and $\mathbf{e}$ the matrix $M_{\mathbf{e}\to \mathbf{a}}$ can encode many familiar doubly-indexed combinatorial sequences, such as binomial coefficients, Stirling numbers of both kinds, Lah numbers and central factorial numbers. In all four of these examples, $M_{\mathbf{e}\to \mathbf{a}}$ enjoys the property of total non-negativity – the determinants of all its square submatrices are non-negative. This leads to a natural question: when, in general, is $M_{\mathbf{e}\to \mathbf{a}}$ totally non-negative? Galvin and Pacurar found a simple condition on $\mathbf{e}$ that characterizes total non-negativity of $M_{\mathbf{e}\to \mathbf{a}}$ when $\mathbf{a}$ is non-decreasing. Here we fully extend this result. For arbitrary real sequences $\mathbf{a}$ and $\mathbf{e}$, we give a condition that can be checked in $O(n^2)$ time that determines whether $M_{\mathbf{e}\to \mathbf{a}}$ is totally non-negative. When $M_{\mathbf{e}\to \mathbf{a}}$ is totally non-negative, we witness this with a planar network whose weights are non-negative and whose path matrix is $M_{\mathbf{e}\to \mathbf{a}}$. When it is not, we witness this with an explicit negative minor.