Definition 9 ($ij$-compression). Fix $1\le i<j\le n$. The $ij$-compression $C_{ij}$ is defined as follows:

For $A\in X^{(r)}$, set

$$C_{ij}(A)=\{\begin{array}{cc}A\cup i-j\hfill & \text{if }j\in A\text{, }i\notin A\hfill \\ A\hfill & \text{otherwise}\hfill \end{array},$$

and for $A\subset X^{(r)}$, set

$$C_{ij}(A)=\{C_{ij}(A):A\in A\}\cup \{A\in A:C_{ij}(A)\in A\}.$$