*3.2. Wall Restrictions*

**Definition 1.** *For lattice paths that start at* (0, 0) *we will say that* W*<sup>L</sup> <sup>d</sup> with d* ≤ 0 *is a left wall restriction (relative to x* = 0*) if at points* (*d*, *y*) *paths are allowed to take steps of type* S*<sup>R</sup> only*

$$\mathcal{W}\_d^L = \{ (d, y) \rightarrow (d + 1, y + 1) \, \
only \}.$$

**Lemma 2.** *The number of paths from* (0, 0) *to* (*M*, *N*) *with the set of steps* S *and one wall restriction* W*<sup>L</sup> <sup>a</sup> can be expressed via the number of unrestricted paths as*

$$|L((0,0)\rightarrow(M,N)\mid\mathcal{W}\_a^L)| = \binom{N}{\frac{N-M}{2}} - \binom{N}{\frac{N-M}{2}+a-1}, \quad \text{for} \quad M \ge a,\tag{2}$$

We considered the left walls located at *x* = 0. An example of possible steps for paths descending from (0, 0) in the presence of this restriction is given in Figure 2.

**Figure 2.** Arrangement of steps for points of L in presence of restriction W*<sup>L</sup>* 0 .

#### *3.3. Filter Restrictions*

**Definition 2.** *For n* ∈ N*, we say that there is a filter* F*<sup>n</sup> <sup>d</sup> of type n, located at x* = *d if at x* = *d*, *d* + 1 *only the following steps are allowed:*

F*<sup>n</sup> <sup>d</sup>* <sup>=</sup> {(*d*, *<sup>y</sup>*) *<sup>n</sup>* −→ (*d* + 1, *y* + 1), (*d* + 1, *y* + 1) → (*d* + 2, *y* + 2), (*d* + 1, *y* + 1) <sup>2</sup> −→ (*d*, *y* + 2)}.

*The index above the arrow is the weight of the step.*

Note that by default, an arrow with no number at the top means that the corresponding step has a weight of 1. An example of possible steps for descending paths in the presence of this restriction is given in Figure 3. We highlighted steps of weight 2 with red instead of an arrow with a superscript 2 for future convenience, as those are the most common for the auxiliary lattice path model and its modifications. We were mostly involved with filters of type 1, so superscripts *n* were avoided, leaving Bratteli diagrams with black and red arrows, with weights 1 and 2 correspondingly.

**Figure 3.** Filter F*<sup>n</sup> <sup>d</sup>* . Red arrows correspond to steps (*<sup>d</sup>* <sup>+</sup> 1, *<sup>y</sup>* <sup>+</sup> <sup>1</sup>) <sup>2</sup> −→ (*d*, *y* + 2) that has a weight 2. Black arrows with superscript *n* correspond to steps (*d*, *y*) *<sup>n</sup>* −→ (*d* + 1, *y* + 1). Other steps have weight 1.

**Lemma 3.** *The number of lattice paths from* (0, 0) *to* (*M*, *N*) *with steps from* S *and filter restriction* F*<sup>n</sup> <sup>d</sup> with x* = *d* > 0 *and n* ∈ N *is*

$$Z(L\_N((0,0)\to(M,N)\mid\mathcal{F}\_d^n)) = \binom{N}{\frac{N-M}{2}} - \binom{N}{\frac{N-M}{2}+d},\text{ for }M$$

$$Z(L\_N((0,0)\to(M,N)\mid\mathcal{F}\_d^n))=n\binom{N}{\left(\frac{N-M}{2}\right)},\text{ for }M>d.\tag{4}$$

**Proof.** The proof is the same as for Lemma 4.8 and Lemma 4.9 in [13].
