for walks ending on the hyperplane $z_k=0$ by setting $z_k=0$ in the series $F({\ensuremath{\mathbf{z}}},t)$ (the variables $z_1,\dots,z_d$ are referred to as *catalytic* variables in the literature, as they are present during the analysis and removed at the end of the ‘reaction’ via specialization to 1). We also define the function (known as either the *characteristic polynomial* or the *inventory* of ${\ensuremath{\mathcal{S}}}$) by $$S({\ensuremath{\mathbf{z}}})= \sum_{{\ensuremath{\mathbf{i}}}\in {\ensuremath{\mathcal{S}}}} {\ensuremath{\mathbf{z}}}^{{\ensuremath{\mathbf{i}}}} = [t^1] F({\ensuremath{\mathbf{z}}},t) \in \mathbb{Q}\left[z_1,{\ensuremath{\overline{z}}}_1,\dots,z_d,{\ensuremath{\overline{z}}}_d\right].$$ In many recent analyses of lattice walks, functional equations are derived by translating the following description of a walk into a generating function equation: a walk is either an empty walk, or a shorter walk followed by a single step. To ensure the condition that the walks remain in the positive orthant, we must not count walks that add a step with a negative $k$-th component to a walk ending on the hyperplane $z_k=0$. To account for this, it is sufficient to subtract an appropriate multiple of $F$ from the functional equation: $t{\ensuremath{\overline{z}}}_k F(z_1, \dots, z_{k-1}, 0, z_{k+1},\dots, z_d, t)$, however if a given step has several negative components we must use inclusion and exclusion to prevent over compensation.

This can be made explicit. Let ${\ensuremath{\mathcal{S}}}\subseteq
  \{1,0,-1\}^d$ define a 