[proofplan] We iterate the multiplicity-free branching rule for polynomial $\mathfrak{gl}_k$-modules along the chain $\mathfrak{gl}_n\supset\mathfrak{gl}_{n-1}\supset\cdots\supset\mathfrak{gl}_1$. At the step from $\mathfrak{gl}_k$ to $\mathfrak{gl}_{k-1}$, the allowed next highest weights are exactly the integer rows interlacing the previous row, and each occurs once. Thus a complete chain of branching choices is precisely a triangular interlacing array with top row $\lambda$. Since the terminal $\mathfrak{gl}_1$-summands are one-dimensional, summing dimensions over all complete chains gives the stated pattern count. [/proofplan]

[step:Iterate the branching rule and record each highest-weight row]For $1\le k\le n$, write a dominant polynomial $\mathfrak{gl}_k(\mathbb C)$-weight as a row \begin{align*} \mu_k=(\mu_{k,1},\dots,\mu_{k,k})\in\mathbb Z_{\ge 0}^k \end{align*} with $\mu_{k,1}\ge \cdots \ge \mu_{k,k}$. For such a row, let $L^{\mathfrak{gl}_k}(\mu_k)$ denote the irreducible polynomial $\mathfrak{gl}_k(\mathbb C)$-module of highest weight $\mu_k$. Apply [citetheorem:9395] to the restriction of $L^{\mathfrak{gl}_k}(\mu_k)$ from $\mathfrak{gl}_k(\mathbb C)$ to the standard upper-left subalgebra $\mathfrak{gl}_{k-1}(\mathbb C)$. The hypotheses are satisfied because $\mu_k$ is a dominant polynomial highest weight. The theorem gives a multiplicity-free decomposition whose summands are precisely $L^{\mathfrak{gl}_{k-1}}(\mu_{k-1})$, where $\mu_{k-1}=(\mu_{k-1,1},\dots,\mu_{k-1,k-1})\in\mathbb Z_{\ge 0}^{k-1}$ satisfies $\mu_{k,i}\ge \mu_{k-1,i}\ge \mu_{k,i+1}$ for every $1\le i\le k-1$. Starting with $\mu_n=\lambda$, repeat this restriction step for $k=n,n-1,\dots,2$. At each stage, a branching choice is exactly the choice of one interlacing next row.[/step]

[guided]We first make explicit what data are being recorded. For each integer $k$ with $1\le k\le n$, a polynomial dominant highest weight for $\mathfrak{gl}_k(\mathbb C)$ is a weakly decreasing row \begin{align*} \mu_k=(\mu_{k,1},\dots,\mu_{k,k})\in\mathbb Z_{\ge 0}^k. \end{align*} The corresponding irreducible polynomial module is denoted $L^{\mathfrak{gl}_k}(\mu_k)$. Now fix $k$ with $2\le k\le n$ and suppose we have already reached a summand $L^{\mathfrak{gl}_k}(\mu_k)$ in the iterated restriction process. We apply [citetheorem:9395] to the standard inclusion $\mathfrak{gl}_{k-1}(\mathbb C)\subset \mathfrak{gl}_k(\mathbb C)$. The theorem applies because $\mu_k$ is a dominant polynomial highest weight, meaning its entries are nonnegative integers in weakly decreasing order. Its conclusion is that the restriction to $\mathfrak{gl}_{k-1}(\mathbb C)$ is multiplicity-free and that the irreducible summands are exactly those of the form $L^{\mathfrak{gl}_{k-1}}(\mu_{k-1})$, where $\mu_{k-1}=(\mu_{k-1,1},\dots,\mu_{k-1,k-1})\in\mathbb Z_{\ge 0}^{k-1}$ satisfies the interlacing inequalities $\mu_{k,i}\ge \mu_{k-1,i}\ge \mu_{k,i+1}$ for every $1\le i\le k-1$. Thus the branching rule does two things at once. It tells us which next rows are allowed, namely exactly the rows interlacing the current row, and it tells us that no allowed next row appears with multiplicity greater than one. Starting from the top row $\mu_n=\lambda$, we may therefore record an iterated branching path by writing down one row of length $k-1$ below each row of length $k$, for $k=n,n-1,\dots,2$.[/guided]

[step:Identify complete branching paths with Gelfand-Tsetlin patterns] A complete branching path is a sequence of rows $\mu_n,\mu_{n-1},\dots,\mu_1$ with $\mu_n=\lambda$ such that, for every $2\le k\le n$ and every $1\le i\le k-1$, $\mu_{k,i}\ge \mu_{k-1,i}\ge \mu_{k,i+1}$. Define a triangular array $(\lambda_{k,i})_{1\le i\le k\le n}$ by $\lambda_{k,i}:=\mu_{k,i}$ for every $1\le i\le k\le n$. This array has top row $\lambda$ and satisfies exactly the defining interlacing inequalities for a Gelfand-Tsetlin pattern. Conversely, given a Gelfand-Tsetlin pattern $(\lambda_{k,i})_{1\le i\le k\le n}$ with top row $\lambda$, define $\mu_k:=(\lambda_{k,1},\dots,\lambda_{k,k})$ for each $1\le k\le n$. The entries of each row are integers by definition of a Gelfand-Tsetlin pattern. The interlacing inequalities imply both nonnegativity and weak decrease: nonnegativity follows by descending from the nonnegative top row, and weak decrease follows from $\lambda_{k,i}\ge \lambda_{k-1,i}\ge \lambda_{k,i+1}$. Hence each $\mu_k$ is a dominant polynomial $\mathfrak{gl}_k(\mathbb C)$-weight. Therefore [citetheorem:9395] applies at every adjacent pair of rows and shows that each lower row is an allowed summand in the restriction from the row above. These two constructions are inverse to each other, so complete branching paths are in bijection with Gelfand-Tsetlin patterns with top row $\lambda$. [/step]

[step:Show that each terminal summand is one-dimensional] Let $a\in\mathbb Z_{\ge 0}$, and consider the terminal polynomial $\mathfrak{gl}_1(\mathbb C)$-module $L^{\mathfrak{gl}_1}(a)$. Since $\mathfrak{gl}_1(\mathbb C)$ is one-dimensional and abelian, a finite-dimensional $\mathfrak{gl}_1(\mathbb C)$-module is the same as a finite-dimensional complex [vector space](/page/Vector%20Space) equipped with one complex-linear endomorphism. That endomorphism has an eigenvector over $\mathbb C$, and the span of such an eigenvector is a nonzero $\mathfrak{gl}_1(\mathbb C)$-submodule. Irreducibility therefore forces the whole module to equal this one-dimensional span. Hence \begin{align*} \dim_{\mathbb C} L^{\mathfrak{gl}_1}(a)=1. \end{align*} [/step]

[step:Count dimensions by summing over all complete paths] Each application of [citetheorem:9395] is a direct-sum decomposition of the restricted module, and restriction along a Lie subalgebra does not change the underlying complex vector space. Iterating these direct-sum decompositions expresses the underlying vector space of $L^{\mathfrak{gl}_n}(\lambda)$ as a [direct sum](/page/Direct%20Sum) of terminal $\mathfrak{gl}_1(\mathbb C)$-summands, one for each complete branching path. The multiplicity-free part of the branching rule ensures that each path contributes exactly one terminal summand. By the preceding step, every terminal summand has complex dimension $1$. Therefore \begin{align*} \dim_{\mathbb C} L^{\mathfrak{gl}_n}(\lambda) = \#\{\text{complete branching paths from }\lambda\text{ to a }\mathfrak{gl}_1\text{ row}\}. \end{align*} Using the bijection between complete branching paths and Gelfand-Tsetlin patterns with top row $\lambda$, this becomes \begin{align*} \dim_{\mathbb C} L^{\mathfrak{gl}_n}(\lambda) = \#\{\text{Gelfand-Tsetlin patterns with top row }\lambda\}. \end{align*} This proves both the indexing statement and the dimension formula. [/step]