I am reading Bruce Sagan's Combinatorics: The Art of Counting. In $\S$7.2 The Schur Basis of $\mathrm{Sym}$, the author states the formulas involving the Jacobi–Trudi determinants and the Schur functions in Theorem 7.2.3. (on page 217):
Thereom 7.2.3. (Jacobi–Trudi Determinants) Suppose $\lambda = (\lambda_1, \dotsc, \lambda_l)$.
(a) $s_{\lambda} = \det[h_{\lambda_i - i + j}]_{1 \leq i, j \leq l}$.
(b) $s_{\lambda^t} = \det[e_{\lambda_i - i + j}]_{1 \leq i, j \leq l}$.
The author proves the first identity, modulo some steps to be completed by the reader, and gives a hint to prove the second identity along the same lines. I am having trouble with using the given hint. I shall briefly describe the proof given for part (a), so that I can point out where I am stuck in applying the hint for part (b).
Consider the directed acyclic graph with vertex set $\mathbb{Z}^2$ and edges $(a,b) \to (a+1,b)$ and $(a,b) \to (a,b+1)$ for every $(a,b) \in \mathbb{Z}^2$. Edges of the former type shall be called east steps and those of the latter type shall be called north steps. Weight the edges as follows: if $e$ is a north step, define $w(e) = 1$, and if $e$ is an east step, define $w(e) = x_{t}$, where $t$ is "the $y$-coordinate" of the edge $e$.
Let $k \geq 1$. Then, if $A = (u_1, \dotsc, u_l)$ and $B = (v_1, \dotsc, v_l)$ are tuples of vertices where $$ u_i = (1 - i, 1) \quad \text{and} \quad v_i = (\lambda_i - i + 1, k), $$ then the path matrix associated to the pair $(A,B)$ turns out to be the matrix in the RHS of (a) under the limit as $k$ tends to infinity. (Note: the author prefers to talk about infinite paths by taking $v_i = (\lambda_i - i + 1,\infty)$, but I prefer reformulating it using a limit to avoid that notion.)
Now, by the Lindstrõm–Gessel–Viennot Lemma, the determinant of the path matrix equals $\sum_{P \in \mathrm{VD}} \mathrm{sgn}(P) w(P)$, where the sum is taken over all vertex-disjoint path systems $P = (P_1, \dotsc, P_l)$ from $A$ to $B$. This means that there is a permutation $\pi \in \mathfrak{S}_l$ associated to $P$ so that each $P_i$ is a path from $u_i$ to $v_{\pi(i)}$, and to say that $P$ is "vertex-disjoint" means that no two paths in $P$ share a vertex. We also have, by definition, that $\mathrm{sgn}(P) = \mathrm{sgn}(\pi)$ and $w(P) = w(P_1) \dotsm w(P_l)$, where the weight of a path is the product of weights of the edges appearing in it. It is an easy verification that if $P \in \mathrm{VD}$, then the associated permutation must be the identity, so $\mathrm{sgn}(P) = 1$ for all $P \in \mathrm{VD}$.
Now, let $\mathrm{SSYT}(\lambda,k)$ be the set of semistandard Young tableau of shape $\lambda$ with entries in $[k]$. Define the map $f \colon \mathrm{VD} \to \mathrm{SSYT}(\lambda,k)$ as follows. Let $P = (P_1,\dotsc,P_l) \in \mathrm{VD}$. For each $P_i$, consider the sequence of $y$-coordinates of its east steps, taken in the same order. Let the $i$th row of the tableau $f(P)$ be this sequence. Then, one can verify that $f(P)$ is indeed a semistandard Young tableau of shape $\lambda$ with entries in $[k]$. The map $f$ also turns out to be a bijection. The weight of a tableau $T$ is defined as $\prod_{(i,j)} x_{T_{ij}}$, where $T_{ij}$ is the $(i,j)$th entry in $T$. So, we see that $f$ is in fact weight-preserving. Thus, $\sum_{P \in \mathrm{VD}} w(P) = \sum_{T \in \mathrm{SSYT}(\lambda,k)} w(T)$, which is the Schur function associated to $\lambda$ in $k$-variables. Letting $k$ tend to infinity, we get the LHS of (a), as required.
Now, the hint given for the second part is as follows: instead of weighting the edges globally, we shall assign a weight to an edge $e$ relative to a path $P$ containing it, as follows. Let $P = (e_1, e_2, \dotsc, e_r)$ be a path. Then, we define $w_P(e_i) = 1$ if $e_i$ is a north step, and $w_P(e_i) = i$ if $e_i$ is an east step. We define the weight of a path $P$ to be $\prod_{e \in P} w_P(e)$. The other definitions remain the same.
The problem I am facing is that the proof of the LGV Lemma is not valid when we take this path-dependent weighting of the edges. To prove the LGV Lemma, we define an involution $\Omega$ on the set of all "intersecting" path systems from $A$ to $B$, such that $\Omega$ is sign-reversing and weight-preserving. However, $\Omega$ fails to be weight-preserving if we take the weights on the edges to be path-dependent.
So, for this reason, I'm not sure how to modify the proof of part (a) to make this hint work.