I have the following set of equation systems, and I would like to find a short, formal way to write it down:

n=4: $$\omega_{A,B,a,b}\cdot\omega_{C,D,c,d}+\omega_{A,C,a,c}\cdot\omega_{B,D,b,d}+\omega_{A,D,a,d}\cdot\omega_{B,C,b,}=\delta_{a,b}\cdot\delta_{b,c}\cdot\delta_{c,d}$$ with $\omega_{X,Y,x,y} \in \mathbb{C}$, and $x,y \in \{0,1\}$. This leads to 24 independent variables ($\omega_{A,B,0,0}$, $\omega_{A,B,0,1}$, $\omega_{A,B,1,0}$, $\omega_{A,B,1,1}$, $\omega_{A,C,0,0}$, $\omega_{A,C,0,1}$, $\omega_{A,C,1,0}$, $\omega_{A,C,1,1}$, $\omega_{A,D,0,0}$, $\omega_{A,D,0,1}$, $\omega_{A,D,1,0}$, $\omega_{A,D,1,1}$, $\omega_{B,C,0,0}$, $\omega_{B,C,0,1}$, $\omega_{B,C,1,0}$, $\omega_{B,C,1,1}$, $\omega_{B,D,0,0}$, $\omega_{B,D,0,1}$, $\omega_{B,D,1,0}$, $\omega_{B,D,1,1}$, $\omega_{C,D,0,0}$, $\omega_{C,D,0,1}$, $\omega_{C,D,1,0}$, $\omega_{C,D,1,1}$), and 16 equations:

$$ \omega_{A,B,0,0}\cdot\omega_{C,D,0,0}+\omega_{A,C,0,0}\cdot\omega_{B,D,0,0}+\omega_{A,D,0,0}\cdot\omega_{B,C,0,0}=1\\ \omega_{A,B,0,0}\cdot\omega_{C,D,0,1}+\omega_{A,C,0,0}\cdot\omega_{B,D,0,1}+\omega_{A,D,0,1}\cdot\omega_{B,C,0,0}=0\\ \omega_{A,B,0,0}\cdot\omega_{C,D,1,0}+\omega_{A,C,0,1}\cdot\omega_{B,D,0,0}+\omega_{A,D,0,0}\cdot\omega_{B,C,0,1}=0\\ \omega_{A,B,0,0}\cdot\omega_{C,D,1,1}+\omega_{A,C,0,1}\cdot\omega_{B,D,0,1}+\omega_{A,D,0,1}\cdot\omega_{B,C,0,1}=0\\ $$ $$ \omega_{A,B,0,1}\cdot\omega_{C,D,0,0}+\omega_{A,C,0,0}\cdot\omega_{B,D,1,0}+\omega_{A,D,0,0}\cdot\omega_{B,C,1,0}=0\\ \omega_{A,B,0,1}\cdot\omega_{C,D,0,1}+\omega_{A,C,0,0}\cdot\omega_{B,D,1,1}+\omega_{A,D,0,1}\cdot\omega_{B,C,1,0}=0\\ \omega_{A,B,0,1}\cdot\omega_{C,D,1,0}+\omega_{A,C,0,1}\cdot\omega_{B,D,1,0}+\omega_{A,D,0,0}\cdot\omega_{B,C,1,1}=0\\ \omega_{A,B,0,1}\cdot\omega_{C,D,1,1}+\omega_{A,C,0,1}\cdot\omega_{B,D,1,1}+\omega_{A,D,0,1}\cdot\omega_{B,C,1,1}=0\\ $$ $$ \omega_{A,B,1,0}\cdot\omega_{C,D,0,0}+\omega_{A,C,1,0}\cdot\omega_{B,D,0,0}+\omega_{A,D,1,0}\cdot\omega_{B,C,0,0}=0\\ \omega_{A,B,1,0}\cdot\omega_{C,D,0,1}+\omega_{A,C,1,0}\cdot\omega_{B,D,0,1}+\omega_{A,D,1,1}\cdot\omega_{B,C,0,0}=0\\ \omega_{A,B,1,0}\cdot\omega_{C,D,1,0}+\omega_{A,C,1,1}\cdot\omega_{B,D,0,0}+\omega_{A,D,1,0}\cdot\omega_{B,C,0,1}=0\\ \omega_{A,B,1,0}\cdot\omega_{C,D,1,1}+\omega_{A,C,1,1}\cdot\omega_{B,D,0,1}+\omega_{A,D,1,1}\cdot\omega_{B,C,0,1}=0\\ $$ $$ \omega_{A,B,1,1}\cdot\omega_{C,D,0,0}+\omega_{A,C,1,0}\cdot\omega_{B,D,1,0}+\omega_{A,D,1,0}\cdot\omega_{B,C,1,0}=0\\ \omega_{A,B,1,1}\cdot\omega_{C,D,0,1}+\omega_{A,C,1,0}\cdot\omega_{B,D,1,1}+\omega_{A,D,1,1}\cdot\omega_{B,C,1,0}=0\\ \omega_{A,B,1,1}\cdot\omega_{C,D,1,0}+\omega_{A,C,1,1}\cdot\omega_{B,D,1,0}+\omega_{A,D,1,0}\cdot\omega_{B,C,1,1}=0\\ \omega_{A,B,1,1}\cdot\omega_{C,D,1,1}+\omega_{A,C,1,1}\cdot\omega_{B,D,1,1}+\omega_{A,D,1,1}\cdot\omega_{B,C,1,1}=1 $$

n=6: $$\omega_{A,B,a,b}\cdot\omega_{C,D,c,d}\cdot\omega_{E,F,e,f}+\omega_{A,B,a,b}\cdot\omega_{C,E,c,e}\cdot\omega_{D,F,d,f}+\omega_{A,B,a,b}\cdot\omega_{C,F,c,f}\cdot\omega_{D,E,d,e}\\ +\omega_{A,C,a,c}\cdot\omega_{B,D,b,d}\cdot\omega_{E,F,e,f}+\omega_{A,C,a,c}\cdot\omega_{B,E,b,e}\cdot\omega_{D,F,d,f}+\omega_{A,C,a,c}\cdot\omega_{B,F,b,f}\cdot\omega_{D,E,d,e}\\ +\omega_{A,D,a,d}\cdot\omega_{B,C,b,c}\cdot\omega_{E,F,e,f}+\omega_{A,D,a,d}\cdot\omega_{B,E,b,e}\cdot\omega_{C,F,c,f}+\omega_{A,D,a,d}\cdot\omega_{B,F,b,f}\cdot\omega_{C,E,c,e}\\ +\omega_{A,E,a,e}\cdot\omega_{B,C,b,c}\cdot\omega_{D,F,d,f}+\omega_{A,E,a,e}\cdot\omega_{B,D,b,d}\cdot\omega_{C,F,c,f}+\omega_{A,E,a,e}\cdot\omega_{B,F,b,f}\cdot\omega_{C,D,c,d}\\ +\omega_{A,F,a,f}\cdot\omega_{B,C,b,c}\cdot\omega_{D,E,d,e}+\omega_{A,F,a,f}\cdot\omega_{B,D,b,d}\cdot\omega_{C,E,c,e}+\omega_{A,F,a,f}\cdot\omega_{B,E,b,e}\cdot\omega_{C,D,c,d}=\delta_{a,b}\cdot\delta_{b,c}\cdot\delta_{c,d}\cdot\delta_{d,e}\cdot\delta_{e,f} $$

• The rule is, that I multiply $m=\left(\frac{n}{2}\right)$ variables $\omega_{X_i,Y_i,x_i,y_i}$ ($\omega_{X_1,Y_1,x_1,y_1}\cdot\omega_{X_2,Y_2,x_2,y_2}\cdot\dots\cdot\omega_{X_m,Y_m,x_m,y_m}$), such that $X_0,Y_0,X_1,Y_1,\dots,X_m,Y_m$ contains each of the first $n$ letters in the alphabet exactly once.

• There are $|\omega|=4\frac{n(n-1)}{2}$ variables and $|Q|=2^n$ equations.

• The indices could be generalized to $x,y \in \{0,1,...,c-1\}$. Then $|\omega|=c^2\frac{n(n-1)}{2}$, $|Q|=c^n$.

Question. How can one write this infinite set of equation systems in a concise, formal way?

PS: The current question is a reformulation of this question on graph theory, independent of graphs and perfect matchings.