[proofplan] We first prove the identity after restricting to a finite alphabet $x_1,\dots,x_n$, where each complete homogeneous symmetric function is represented by weighted lattice paths. The determinant is expanded as a signed generating function over all path families from prescribed sources to prescribed sinks. A sign-reversing, weight-preserving tail-swap involution cancels every intersecting family, leaving only nonintersecting families with the identity matching. Finally, nonintersecting path families are identified weight-preservingly with semistandard tableaux of skew shape $\lambda/\mu$, and the finite-alphabet identities stabilize to the symmetric-function identity. [/proofplan]

[step:Reduce to finite alphabets] For $n \in \mathbb{N}$, let $S_n$ denote the symmetric group on the set $\{1,\dots,n\}$, acting on $\mathbb{Z}[x_1,\dots,x_n]$ by permuting variables. Let $\Lambda_n = \mathbb{Z}[x_1,\dots,x_n]^{S_n}$, and let $h_m^{(n)} \in \Lambda_n$ denote the complete homogeneous symmetric polynomial \begin{align*} h_m^{(n)} = \sum_{1 \leq a_1 \leq \cdots \leq a_m \leq n} x_{a_1}\cdots x_{a_m} \end{align*} for $m \geq 1$, with $h_0^{(n)} = 1$ and $h_m^{(n)} = 0$ for $m < 0$. Let $s_{\lambda/\mu}^{(n)}$ denote the finite-alphabet skew Schur polynomial \begin{align*} s_{\lambda/\mu}^{(n)} = \sum_T \prod_{(i,c) \in \lambda/\mu} x_{T(i,c)}, \end{align*} where the sum is over all semistandard tableaux $T: \lambda/\mu \to \{1,\dots,n\}$, meaning $T(i,c) \leq T(i,c+1)$ along rows and $T(i,c) < T(i+1,c)$ down columns whenever the indicated cells belong to $\lambda/\mu$. It is enough to prove \begin{align*} s_{\lambda/\mu}^{(n)} = \det\left(h_{\lambda_i-\mu_j-i+j}^{(n)}\right)_{1 \leq i,j \leq \ell} \end{align*} for every $n \in \mathbb{N}$. Indeed, the symmetric function $h_m$ is the stable limit of $h_m^{(n)}$, and $s_{\lambda/\mu}$ is the stable limit of $s_{\lambda/\mu}^{(n)}$ in each homogeneous degree. [/step]

[step:Encode the complete homogeneous functions by weighted paths] Fix $n \in \mathbb{N}$. Define a directed weighted lattice graph $G_n$ as follows. Its vertex set is \begin{align*} V_n = \mathbb{Z} \times \{1,\dots,n\}. \end{align*} For each $(a,k) \in \mathbb{Z}\times\{1,\dots,n\}$ with $k<n$, include a vertical edge \begin{align*} (a,k) \to (a,k+1) \end{align*} of weight $1$. For each $(a,k) \in V_n$, include a horizontal edge \begin{align*} (a,k) \to (a+1,k) \end{align*} of weight $x_k$. The weight of a directed path is the product of the weights of its edges. For $1 \leq i,j \leq \ell$, define the source and sink vertices \begin{align*} A_j &= (\mu_j-j,1), \\ B_i &= (\lambda_i-i,n). \end{align*} A path from $A_j$ to $B_i$ exists exactly when $\lambda_i-\mu_j-i+j \geq 0$. Such a path has exactly $\lambda_i-\mu_j-i+j$ horizontal edges. Reading the levels of these horizontal edges from left to right gives a weakly increasing sequence \begin{align*} 1 \leq a_1 \leq \cdots \leq a_{\lambda_i-\mu_j-i+j} \leq n, \end{align*} and its weight is $x_{a_1}\cdots x_{a_{\lambda_i-\mu_j-i+j}}$. Conversely, every such weakly increasing sequence determines a unique path by placing the horizontal edges at those levels and using vertical edges between them. Therefore the total weight of all paths from $A_j$ to $B_i$ is \begin{align*} h_{\lambda_i-\mu_j-i+j}^{(n)}. \end{align*} [/step]

[step:Cancel intersecting path families by a sign-reversing tail swap]Expanding the determinant gives \begin{align*} \det\left(h_{\lambda_i-\mu_j-i+j}^{(n)}\right)_{1 \leq i,j \leq \ell} = \sum_{\pi \in S_\ell} \operatorname{sgn}(\pi) \sum_{\mathcal{P} \in \mathcal{F}_\pi} \operatorname{wt}(\mathcal{P}), \end{align*} where $\mathcal{F}_\pi$ is the set of $\ell$-tuples $\mathcal{P}=(P_1,\dots,P_\ell)$ such that $P_j$ is a directed path from $A_j$ to $B_{\pi(j)}$, and \begin{align*} \operatorname{wt}(\mathcal{P}) = \prod_{j=1}^{\ell}\operatorname{wt}(P_j). \end{align*} This is the determinant expansion in the form \begin{align*} \det(M_{ij})_{1 \leq i,j \leq \ell} = \sum_{\pi \in S_\ell} \operatorname{sgn}(\pi)\prod_{j=1}^{\ell} M_{\pi(j),j}, \end{align*} with $M_{ij}=h_{\lambda_i-\mu_j-i+j}^{(n)}$. [claim:Intersecting families cancel in pairs] The total signed weight of all path families in which two paths share a vertex is zero. [/claim] [proof] Given an intersecting family $\mathcal{P}=(P_1,\dots,P_\ell) \in \mathcal{F}_\pi$, choose the smallest sink index $r$ such that $P_r$ intersects some $P_s$ with $s>r$. Along $P_r$, choose the first vertex $v$ that lies on another path $P_s$, and among such paths choose the smallest $s>r$ whose path contains $v$. Define a new family $\mathcal{P}'$ by leaving all paths except $P_r$ and $P_s$ unchanged, and by swapping the terminal tails of $P_r$ and $P_s$ after the common vertex $v$. More explicitly, write \begin{align*} P_r &= P_r^{\mathrm{pre}} \cdot P_r^{\mathrm{tail}}, \\ P_s &= P_s^{\mathrm{pre}} \cdot P_s^{\mathrm{tail}}, \end{align*} where $P_r^{\mathrm{pre}}$ and $P_s^{\mathrm{pre}}$ end at $v$, while $P_r^{\mathrm{tail}}$ and $P_s^{\mathrm{tail}}$ begin at $v$. Then set \begin{align*} P_r' &= P_r^{\mathrm{pre}} \cdot P_s^{\mathrm{tail}}, \\ P_s' &= P_s^{\mathrm{pre}} \cdot P_r^{\mathrm{tail}}. \end{align*} The new path $P_r'$ starts at $A_r$ and ends at $B_{\pi(s)}$, while the new path $P_s'$ starts at $A_s$ and ends at $B_{\pi(r)}$. Hence the associated permutation is $\pi \circ (r\,s)$, where $(r\,s)$ transposes the two source indices. Since $\operatorname{sgn}(\pi \circ (r\,s))=-\operatorname{sgn}(\pi)$, the sign changes by a factor of $-1$. The product of all edge weights is unchanged, because the same directed edge segments occur in $\mathcal{P}'$ as in $\mathcal{P}$, only assigned to the two paths differently. Applying the same construction to $\mathcal{P}'$ selects the same common vertex $v$ and swaps the same two tails back. Thus this map is a weight-preserving involution that reverses the permutation sign. The signed contributions of intersecting families cancel pairwise. [/proof][/step]

[guided]We now explain the cancellation mechanism in full, since this is the determinant argument that drives the proof. The determinant expansion is a signed sum indexed by permutations: \begin{align*} \det\left(h_{\lambda_i-\mu_j-i+j}^{(n)}\right)_{1 \leq i,j \leq \ell} = \sum_{\pi \in S_\ell} \operatorname{sgn}(\pi) \sum_{\mathcal{P} \in \mathcal{F}_\pi} \operatorname{wt}(\mathcal{P}), \end{align*} where $\mathcal{F}_\pi$ consists of path families $\mathcal{P}=(P_1,\dots,P_\ell)$ with $P_j$ running from $A_j$ to $B_{\pi(j)}$. The sign records the matching from source indices to sink indices. Suppose two paths in $\mathcal{P}$ meet. To pair this family with another family of the opposite sign, choose the smallest index $r$ such that $P_r$ meets some later path. Along $P_r$, choose the first vertex $v$ that lies on another path, and choose the smallest $s>r$ such that $v \in P_s$. Since the graph is directed, each path decomposes uniquely into an initial segment ending at $v$ and a terminal segment starting at $v$: \begin{align*} P_r &= P_r^{\mathrm{pre}} \cdot P_r^{\mathrm{tail}}, \\ P_s &= P_s^{\mathrm{pre}} \cdot P_s^{\mathrm{tail}}. \end{align*} Now swap the tails: \begin{align*} P_r' &= P_r^{\mathrm{pre}} \cdot P_s^{\mathrm{tail}}, \\ P_s' &= P_s^{\mathrm{pre}} \cdot P_r^{\mathrm{tail}}. \end{align*} These are still directed paths because both tails begin at the same vertex $v$. The labels of the paths remain attached to their sources: $P_r'$ still starts at $A_r$ and $P_s'$ still starts at $A_s$. What changes is the sink assignment. If $P_r$ originally ended at $B_{\pi(r)}$ and $P_s$ originally ended at $B_{\pi(s)}$, then $P_r'$ ends at $B_{\pi(s)}$ and $P_s'$ ends at $B_{\pi(r)}$. Thus the new source-to-sink permutation is $\pi \circ (r\,s)$, not an ill-defined relabelling of the tuple. Since a transposition has sign $-1$, we have \begin{align*} \operatorname{sgn}(\pi \circ (r\,s))=-\operatorname{sgn}(\pi). \end{align*} The weight is unchanged. Indeed, the edge multiset appearing in $P_r \cup P_s$ is the same as the edge multiset appearing in $P_r' \cup P_s'$, and every other path is left fixed. Hence \begin{align*} \operatorname{wt}(\mathcal{P}') = \operatorname{wt}(\mathcal{P}). \end{align*} Finally, the rule is an involution: applying it again to $\mathcal{P}'$ finds the same first intersection vertex $v$ and swaps the same two tails back. Thus every intersecting family is paired with exactly one family of the same weight and opposite sign, so the total signed contribution of intersecting families is $0$.[/guided]

[step:Identify the surviving families with semistandard skew tableaux] After the cancellation, only vertex-disjoint path families remain. The strict inequalities \begin{align*} \mu_1-1 > \mu_2-2 > \cdots > \mu_\ell-\ell \end{align*} and \begin{align*} \lambda_1-1 > \lambda_2-2 > \cdots > \lambda_\ell-\ell \end{align*} show that a nonintersecting family can only connect $A_i$ to $B_i$ for every $i$. Hence the surviving signed sum has positive sign and equals the total weight of all nonintersecting families $(P_1,\dots,P_\ell)$ with $P_i:A_i\to B_i$. Given such a nonintersecting family, define a filling \begin{align*} T_{\mathcal{P}}: \lambda/\mu \to \{1,\dots,n\} \end{align*} as follows. For a cell $(i,c)$ of the skew diagram, where $1 \leq i \leq \ell$ and $\mu_i < c \leq \lambda_i$, set $T_{\mathcal{P}}(i,c)=k$ if the path $P_i$ contains the horizontal edge \begin{align*} (c-i-1,k) \to (c-i,k). \end{align*} Each such edge exists uniquely, since $P_i$ moves from horizontal coordinate $\mu_i-i$ to horizontal coordinate $\lambda_i-i$ by one-unit horizontal steps. Along a fixed row $i$, the levels of the horizontal edges of $P_i$ weakly increase from left to right, because the only vertical edges move from level $k$ to level $k+1$. Hence $T_{\mathcal{P}}(i,c) \leq T_{\mathcal{P}}(i,c+1)$. Now suppose cells $(i,c)$ and $(i+1,c)$ both lie in $\lambda/\mu$. The corresponding horizontal edge of $P_i$ lies one horizontal unit to the right of the corresponding horizontal edge of $P_{i+1}$. If their levels satisfied \begin{align*} T_{\mathcal{P}}(i,c) \geq T_{\mathcal{P}}(i+1,c), \end{align*} then the upward directed path $P_{i+1}$ would have to reach or cross the vertex used by $P_i$ before $P_i$ moves past it, forcing a common vertex. This contradicts nonintersection. Therefore \begin{align*} T_{\mathcal{P}}(i,c) < T_{\mathcal{P}}(i+1,c). \end{align*} Thus $T_{\mathcal{P}}$ is semistandard. Conversely, given a semistandard tableau \begin{align*} T: \lambda/\mu \to \{1,\dots,n\}, \end{align*} construct $P_i$ by placing, for each cell $(i,c)$, the horizontal edge \begin{align*} (c-i-1,T(i,c)) \to (c-i,T(i,c)), \end{align*} and connecting consecutive horizontal edges by vertical edges. The row weak increase condition makes this a directed path from $A_i$ to $B_i$. The column strict increase condition prevents two constructed paths from sharing a vertex. Hence this construction is inverse to $\mathcal{P} \mapsto T_{\mathcal{P}}$. The bijection preserves weights, since \begin{align*} \operatorname{wt}(\mathcal{P}) = \prod_{i=1}^{\ell}\prod_{\mu_i<c\leq\lambda_i} x_{T_{\mathcal{P}}(i,c)}. \end{align*} Therefore the surviving path generating function is $s_{\lambda/\mu}^{(n)}$. [/step]

[step:Pass from finite alphabets to symmetric functions] For every $n \in \mathbb{N}$, the preceding steps prove \begin{align*} s_{\lambda/\mu}^{(n)} = \det\left(h_{\lambda_i-\mu_j-i+j}^{(n)}\right)_{1 \leq i,j \leq \ell}. \end{align*} The skew shape $\lambda/\mu$ has finite size, so both sides are homogeneous of degree $|\lambda|-|\mu|$. Taking the stable limit as $n \to \infty$ sends $s_{\lambda/\mu}^{(n)}$ to $s_{\lambda/\mu}$ and sends each $h_m^{(n)}$ to $h_m$. Since determinants are finite polynomial expressions in their entries, the limit gives \begin{align*} s_{\lambda/\mu} = \det\left(h_{\lambda_i-\mu_j-i+j}\right)_{1 \leq i,j \leq \ell}. \end{align*} This is the desired skew [Jacobi-Trudi identity](/theorems/5181). [/step]