function {#sec:lwalks}
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Fix a dimension $d$ and a highly symmetric set of steps ${\ensuremath{\mathcal{S}}}\subseteq
\{\pm1,0\}^d \setminus \{\mathbf{0}\}$. Recall this means that $(i_1,\dots,i_k,\dots,i_d) \in {\ensuremath{\mathcal{S}}}$ implies $(i_1,\dots,-i_k,\dots,i_d) \in {\ensuremath{\mathcal{S}}}$. In this section we derive a functional equation for a multivariate generating function, apply the orbit sum method to derive a closed expression related to this generating function, and conclude by writing the univariate counting generating function for the number of walks as the complete diagonal of a rational function.

The following notation is used throughout: $${\ensuremath{\overline{z}}}_i= z_i^{-1}; \qquad {\ensuremath{\mathbf{z}}}= (z_1, \dots, z_d); \qquad {\ensuremath{\mathbf{i}}}=( i_1,i_2,\dots,i_d)\in \mathbb{Z}^d; \qquad {\ensuremath{\mathbf{z}}}^{{\ensuremath{\mathbf{i}}}} = z_1^{i_1}\cdots z_d^{i_d},$$ and we write $\mathbb{Q}[z_k,\overline{z_k}]$ to refer to the ring of Laurent polynomials in the variable $z_k$.

A functional equation
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To begin, we define the generating function: $$F({\ensuremath{\mathbf{z}}},t)= \sum_{\substack{n \geq 0 \\ {\ensuremath{\mathbf{i}}}\in \mathbb{Z}^d}} s_{{\ensuremath{\mathbf{i}}}}(n) {\ensuremath{\mathbf{z}}}^{{\ensuremath{\mathbf{i}}}} t^n 
  = \sum_{n \geq 0} \left(\sum_{{\ensuremath{\mathbf{i}}}\in \mathbb{Z}^d}s_{{\ensuremath{\mathbf{i}}}}(n) z_1^{i_1}\cdots z_d^{i_d}\right) t^n \in \mathbb{Q}[z_1,{\ensuremath{\overline{z}}}_1,\dots,z_d,{\ensuremath{\overline{z}}}_d][\![t]\!],$$ where $s_{{\ensuremath{\mathbf{i}}}}(n)$ counts the number of walks of length $n$ taking steps from ${\ensuremath{\mathcal{S}}}$ which stay in the positive orthant and end at lattice point ${\ensuremath{\mathbf{i}}}\in \mathbb{Z}^d$. Note that the series $F({\ensuremath{\mathbf{1}}},t)$ is the generating function for the total number of walks in the orthant, and we can recover the series 