Crossposting from [MSE][1] after getting no answers. The bounty on the MSE question is still open, but not for long. Be advised that the comments of the MSE question regard an obsolete version, and that the issues which had arose from those comments were answered in subsequent edits.

This question arose in the context of studying reflections.

Definitions, examples and observations
=======================
## Matrix ##
Let $n$ be a positive integer.

Denote by $B_n$ the matrix of dimensions $ 2^n \times \left( n+1 \right) $ with entries from $ \{0,1\} $ such that it satisfies the recursive block relation
$$B_n = 
\left[
\begin{array}{c|c}
\underline{0}_{\left(2^{n-1} \times 1\right)} & B_{n-1}\\
\hline
\underline{1}_{\left(2^{n-1} \times 1\right)} & B_{n-1}
\end{array}
\right] 
$$

with the condition 

$$
B_1 \equiv
\begin{bmatrix}
0 & 0 \\
1 & 0 \\
\end{bmatrix}
$$
### Matrix examples ###
For $ n \in \{2,3,4\} $ obtain 
$$
B_2 =
\begin{bmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
1 & 0 & 0 \\
1 & 1 & 0 \\
\end{bmatrix},
\,
B_3 =
\begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 \\
1 & 1 & 1 & 0 \\
\end{bmatrix},
\,
B_4 =
\begin{bmatrix}
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 \\
1 & 1 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 0 \\
1 & 1 & 1 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 \\
\end{bmatrix}
$$

### Explicit formula for the matrix elements ###
It's not hard to show that
$$
\left(B_n\right)_{i,j} =
\begin{cases}
\lfloor {i-1 \over 2^{n-j}} \rfloor \pmod{2},  & \text{if $1 \le j \le n$}
\\
0, & \text{if $j=n+1$}
\end{cases}
$$
## Path ##
A $B_n$-path $P$ is a set of size $2^n$ where each element is an ordered pair, where the first element is a row index of $B_n$, and the second element is a column index of $B_n$, so that each row index of $B_n$ appears exactly **once** in the elements of $P$. 

Notice that $P$ has the form
$$
\{ 
\left(i_1,j_1\right),\left(i_2,j_2\right), \ldots , \left(i_{2^n},j_{2^n}\right)
\}
$$
where the row indices from all the pairs are **pairwise distinct**.

In other words, a $B_n$-path is equivalent to choosing exactly one element from each and every row of $B_n$ in some order.

Obviously $\left(B_n \right)_{i_{1},j_{1}} = \left(B_n \right)_{i_{2},j_{2}}$ does **not** imply that $\left(i_1,j_1 \right) = \left(i_2,j_2 \right)$.

## Weighted path ##
A $B_n$-weight $w$ is an $\left(n+1\right)$-tuple with non-negative integer entries, such that the sum of its entries is equal to $2^n$.

Fix a $B_n$-weight $w \equiv \left(\mu_1, \mu_2, \ldots , \mu_{n+1} \right) $, so $\mu_j \in \mathbb{Z}_{\ge 0}, \, j \in \{1,2, \ldots, n+1 \}$ and  $\sum_{j=1}^{n+1}{\mu_j} = 2^n$.

A $B_n$-path with $B_n$-weight $w$, denoted by $P_w$, is a $B_n$-path such that $\mu_1$ of its pair elements have column indices which are equal to $1$, $\mu_2$ of the remaining pair elements have column indices which are equal to $2$, and so on, until finally the remaining $\mu_{n+1}$ pair elements have column indices which are equal to $n+1$. 

Notice that if $\mu_k = 0$ for some $ k \in \{1,2,\ldots,n+1\} $ then $P_w$ does not have an element pair with $k$ as a column index.

Notice that the number of **distinct** $B_n$-paths with a fixed weight $w$ is given by the multinomial coefficient 
$$
\binom{\mu_1+\cdots+\mu_{n+1}}{\mu_1,\ldots,\mu_{n+1}}=\binom{2^n}{\mu_1,\ldots,\mu_{n+1}}
$$
### Weighted path examples ###
Consider the matrix $B_2$ and the $B_2$-weight $w \equiv \left(1,2,1 \right)$. A $B_2$-path with $B_n$-weight $w$, denoted by $P_w$, can be, for instance, the set
$$
\{ 
\left( 1,1\right),\left( 2,2\right),\left( 3,2\right),\left( 4,3\right) 
\}
$$
Graphically, this $B_2$-path looks like the following (in red):
$$
\begin{bmatrix}
\color{red}{0} & 0 & 0 \\
0 & \color{red}{1} & 0 \\
1 & \color{red}{0} & 0 \\
1 & 1 & \color{red}{0} \\
\end{bmatrix}
$$
Another possiblity for $P_w$ is the set 
$$
\{ 
\left( 1,2\right),\left( 2,3\right),\left( 3,2\right),\left( 4,1\right) 
\}
$$
which looks like the following:
$$
\begin{bmatrix}
0 & \color{red}{0} & 0 \\
0 & 1 & \color{red}{0} \\
1 & \color{red}{0} & 0 \\
\color{red}{1} & 1 & 0 \\
\end{bmatrix}
$$
Consider the matrix $B_3$ and the $B_3$-weight $w \equiv \left(2,0,5,1 \right)$. A $B_3$-path with $B_n$-weight $w$, denoted by $P_w$ can be, for instance, the set 
$$
\{ 
\left( 1,1\right),\left( 2,1\right),\left( 3,3\right),\left( 4,3\right),\left( 5,3\right),\left( 6,3\right),\left( 7,3\right),\left( 8,4\right)
\}
$$
Graphically, this $B_3$-path looks like the following (in red):
$$
\begin{bmatrix}
\color{red}{0} & 0 & 0 & 0 \\
\color{red}{0} & 0 & 1 & 0 \\
0 & 1 & \color{red}{0} & 0 \\
0 & 1 & \color{red}{1} & 0 \\
1 & 0 & \color{red}{0} & 0 \\
1 & 0 & \color{red}{1} & 0 \\
1 & 1 & \color{red}{0} & 0 \\
1 & 1 & 1 & \color{red}{0} \\
\end{bmatrix}
$$
Another possiblity for $p_w$ is the set
$$
\left( 
\left( 1,4\right),\left( 2,3\right),\left( 3,1\right),\left( 4,3\right),\left( 5,3\right),\left( 6,3\right),\left( 7,3\right),\left( 8,1\right) 
\right)
$$
which looks like the following:
$$
\begin{bmatrix}
0 & 0 & 0 & \color{red}{0} \\
0 & 0 & \color{red}{1} & 0 \\
\color{red}{0} & 1 & 0 & 0 \\
0 & 1 & \color{red}{1} & 0 \\
1 & 0 & \color{red}{0} & 0 \\
1 & 0 & \color{red}{1} & 0 \\
1 & 1 & \color{red}{0} & 0 \\
\color{red}{1} & 1 & 1 & 0 \\
\end{bmatrix}
$$
## Parity of a path ##
The parity of a $B_n$-path $P$ is the sum modulo $2$ of the elements of $B_n$ with row-column indices which correspond to the elements of $P$. 

Summation modulo 2 is commutative, so the parity of a $B_n$-path $P$ is given by
$$
\sum_{i=1}^{2^n}{\left( B_n\right)_{i,j_i}} \pmod 2
$$
where $j_i$ is the column index in the element pair of $P$ with row index $i$.

Notice that when calculatiing this sum we may ignore the elements of $P$  with column index $j_i=n+1$, because the corresponding elements of $B_n$ are all equal to $0$.

### Parity of a path examples ###
Consider the following $B_2$-path and $B_3$-path and just take the sum of the red colored $0$'s and $1$'s modulo 2. 

The $B_2$-path described graphically by 
$$
\begin{bmatrix}
0 & \color{red}{0} & 0 \\
0 & 1 & \color{red}{0} \\
\color{red}{1} & 0 & 0 \\
1 & 1 & \color{red}{0} \\
\end{bmatrix}
$$
has parity equal to $1$.

The $B_3$-path described graphically by 
$$
\begin{bmatrix}
0 & \color{red}{0} & 0 & 0 \\
0 & \color{red}{0} & 1 & 0 \\
0 & 1 & \color{red}{0} & 0 \\
0 & 1 & \color{red}{1} & 0 \\
1 & 0 & \color{red}{0} & 0 \\
1 & 0 & \color{red}{1} & 0 \\
1 & 1 & \color{red}{0} & 0 \\
1 & 1 & 1 & \color{red}{0} \\
\end{bmatrix}
$$
has parity equal to $0$.
### Parity of a weighted path ###
Consider the matrix $B_n$.

Fix a $B_n$-weight $w \equiv \left(\mu_1, \mu_2, \ldots,\mu_{n+1} \right)$, so $\mu_j \in \mathbb{Z}_{\ge 0}, \, j \in \{1,2, \ldots, n+1\}$ and $\sum_{j=1}^{n+1}{\mu_j} = 2^n$.

The parity of a $B_n$-path with weight $w$ can be computed by doing the following:

Consider the the binary representation of the integers in the range $\left[0,2^{n}-1\right]$. Choose $\mu_1$ of them and do a logic shift right by $1$ on each, then choose $\mu_2$ of the remaining and do a logic shift right by $2$ on each,..., then finally do a logic shift right by $n$ on each of the last remaining $\mu_n$. Now look at the LSB of all the shifted integers, then count how many $1$'s there are. The parity of the count is the parity of the corresponding $B_n$-path with weight $w$, where we identify each integer $i$ in the range with the $i+1$-th row of $B_n$.

# Problems #
Consider the matrix $B_n$.

Fix a $B_n$-weight $w \equiv \left(\mu_1, \mu_2, \ldots,\mu_{n+1} \right)$, so $\mu_j \in \mathbb{Z}_{\ge 0}, \, j \in \{1,2, \ldots, n+1\}$ and $\sum_{j=1}^{n+1}{\mu_j} = 2^n$.

1. Show that the number of all distinct $B_n$-paths with weight $w$ and parity equal to $0$ is equal to the number of all distinct $B_n$-paths with weight $w$ and parity equal to $1$, **if and only if** at least one of the entries of the weight $w$ is an **odd** integer.

Now consider a weight with only **even** entries.

Fix a weight $\varpi \equiv \left(2\phi_1, 2\phi_2, \ldots , 2\phi_{n+1} \right) $, so $\phi_j \in \mathbb{Z}_{\ge 0}, \, j \in \{1,2, \ldots, n+1 \}$ and $\sum_{j=1}^{n+1}{\phi_j} = 2^{n-1}$.

2. Count the number all distinct $B_n$-paths with weight $\varpi$ and parity equal to $0$. Count the same for when the parity is equal to $1$.
3. Show that the **difference** between the number of all distinct $B_n$-paths with weight $\varpi$ and parity equal to $0$, and the number of all distinct $B_n$-paths with weight $\varpi$ and parity equal to $1$, is **invariant** under any permutation of the entries of $\varpi$.
# What I am asking for #
I am looking for references to this kind of problems. I'd appreciate to know about equivalent problems which require less setup, perhaps stated as a problem in graph theory. I am also hoping for some input or hints for these problems. Problem 2 seems to be the most difficult.


  [1]: https://math.stackexchange.com/questions/3813417/lattice-paths-like-problems-over-a-certain-rectangle-binary-matrix