I have the following set of equation systems, and I would like to find a short, formal way to write it down. My main difficulty is that I cannot find a good way to write the indices of the variables $\omega$. Any suggestion is highly apprechiated.
**n=4:**
$$\omega_{A,B,a,b}\cdot\omega_{C,D,c,d}+\omega_{A,C,a,c}\cdot\omega_{B,D,b,d}+\omega_{A,D,a,d}\cdot\omega_{B,C,b,c}=\delta_{a,b}\cdot\delta_{b,c}\cdot\delta_{c,d}$$
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:
$$
\omega_{A,B,0,0}\cdot\omega_{C,D,0,0}+\omega_{A,C,0,0}\cdot\omega_{B,D,0,0}+\omega_{A,D,0,0}\cdot\omega_{B,C,0,0}=1\\
\omega_{A,B,0,0}\cdot\omega_{C,D,0,1}+\omega_{A,C,0,0}\cdot\omega_{B,D,0,1}+\omega_{A,D,0,1}\cdot\omega_{B,C,0,0}=0\\
\omega_{A,B,0,0}\cdot\omega_{C,D,1,0}+\omega_{A,C,0,1}\cdot\omega_{B,D,0,0}+\omega_{A,D,0,0}\cdot\omega_{B,C,0,1}=0\\
\omega_{A,B,0,0}\cdot\omega_{C,D,1,1}+\omega_{A,C,0,1}\cdot\omega_{B,D,0,1}+\omega_{A,D,0,1}\cdot\omega_{B,C,0,1}=0\\
$$
$$
\omega_{A,B,0,1}\cdot\omega_{C,D,0,0}+\omega_{A,C,0,0}\cdot\omega_{B,D,1,0}+\omega_{A,D,0,0}\cdot\omega_{B,C,1,0}=0\\
\omega_{A,B,0,1}\cdot\omega_{C,D,0,1}+\omega_{A,C,0,0}\cdot\omega_{B,D,1,1}+\omega_{A,D,0,1}\cdot\omega_{B,C,1,0}=0\\
\omega_{A,B,0,1}\cdot\omega_{C,D,1,0}+\omega_{A,C,0,1}\cdot\omega_{B,D,1,0}+\omega_{A,D,0,0}\cdot\omega_{B,C,1,1}=0\\
\omega_{A,B,0,1}\cdot\omega_{C,D,1,1}+\omega_{A,C,0,1}\cdot\omega_{B,D,1,1}+\omega_{A,D,0,1}\cdot\omega_{B,C,1,1}=0\\
$$
$$
\omega_{A,B,1,0}\cdot\omega_{C,D,0,0}+\omega_{A,C,1,0}\cdot\omega_{B,D,0,0}+\omega_{A,D,1,0}\cdot\omega_{B,C,0,0}=0\\
\omega_{A,B,1,0}\cdot\omega_{C,D,0,1}+\omega_{A,C,1,0}\cdot\omega_{B,D,0,1}+\omega_{A,D,1,1}\cdot\omega_{B,C,0,0}=0\\
\omega_{A,B,1,0}\cdot\omega_{C,D,1,0}+\omega_{A,C,1,1}\cdot\omega_{B,D,0,0}+\omega_{A,D,1,0}\cdot\omega_{B,C,0,1}=0\\
\omega_{A,B,1,0}\cdot\omega_{C,D,1,1}+\omega_{A,C,1,1}\cdot\omega_{B,D,0,1}+\omega_{A,D,1,1}\cdot\omega_{B,C,0,1}=0\\
$$
$$
\omega_{A,B,1,1}\cdot\omega_{C,D,0,0}+\omega_{A,C,1,0}\cdot\omega_{B,D,1,0}+\omega_{A,D,1,0}\cdot\omega_{B,C,1,0}=0\\
\omega_{A,B,1,1}\cdot\omega_{C,D,0,1}+\omega_{A,C,1,0}\cdot\omega_{B,D,1,1}+\omega_{A,D,1,1}\cdot\omega_{B,C,1,0}=0\\
\omega_{A,B,1,1}\cdot\omega_{C,D,1,0}+\omega_{A,C,1,1}\cdot\omega_{B,D,1,0}+\omega_{A,D,1,0}\cdot\omega_{B,C,1,1}=0\\
\omega_{A,B,1,1}\cdot\omega_{C,D,1,1}+\omega_{A,C,1,1}\cdot\omega_{B,D,1,1}+\omega_{A,D,1,1}\cdot\omega_{B,C,1,1}=1
$$
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**n=6:**
$$\omega_{A,B,a,b}\cdot\omega_{C,D,c,d}\cdot\omega_{E,F,e,f}+\omega_{A,B,a,b}\cdot\omega_{C,E,c,e}\cdot\omega_{D,F,d,f}+\omega_{A,B,a,b}\cdot\omega_{C,F,c,f}\cdot\omega_{D,E,d,e}\\
+\omega_{A,C,a,c}\cdot\omega_{B,D,b,d}\cdot\omega_{E,F,e,f}+\omega_{A,C,a,c}\cdot\omega_{B,E,b,e}\cdot\omega_{D,F,d,f}+\omega_{A,C,a,c}\cdot\omega_{B,F,b,f}\cdot\omega_{D,E,d,e}\\
+\omega_{A,D,a,d}\cdot\omega_{B,C,b,c}\cdot\omega_{E,F,e,f}+\omega_{A,D,a,d}\cdot\omega_{B,E,b,e}\cdot\omega_{C,F,c,f}+\omega_{A,D,a,d}\cdot\omega_{B,F,b,f}\cdot\omega_{C,E,c,e}\\
+\omega_{A,E,a,e}\cdot\omega_{B,C,b,c}\cdot\omega_{D,F,d,f}+\omega_{A,E,a,e}\cdot\omega_{B,D,b,d}\cdot\omega_{C,F,c,f}+\omega_{A,E,a,e}\cdot\omega_{B,F,b,f}\cdot\omega_{C,D,c,d}\\
+\omega_{A,F,a,f}\cdot\omega_{B,C,b,c}\cdot\omega_{D,E,d,e}+\omega_{A,F,a,f}\cdot\omega_{B,D,b,d}\cdot\omega_{C,E,c,e}+\omega_{A,F,a,f}\cdot\omega_{B,E,b,e}\cdot\omega_{C,D,c,d}=\delta_{a,b}\cdot\delta_{b,c}\cdot\delta_{c,d}\cdot\delta_{d,e}\cdot\delta_{e,f}
$$
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- 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 $|\omega|=c^2\frac{n(n-1)}{2}$, $|Q|=c^n$.
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>**Question.** How can one write this infinite set of equation systems in a concise, formal way?
There are infinite equation systems because $n$ can be an arbitrary even integer, and $c$ can contain arbitrarily many terms.
PS: The current question is a reformulation of this question on [graph theory][1], independent of graphs and perfect matchings.
.....
[1]: https://mathoverflow.net/questions/311325/vertex-coloring-inherited-from-perfect-matchings-motivated-by-quantum-physics