0x6892cfdd

Let \(k\ge 1\), \(\Pi_k\) be the set of all pairings of \(\{1,\ldots ,2k\}\) and \(a,b,c\in \{1,\ldots ,2k\}\) be pairwise distinct. Then

\[ \tau_{a,b} = \tau_{a,c}\circ \tau_{b,c} \circ \tau_{a,c}. \]

Note

We can use the group action of \(S_{2k}\) on \(\Pi_k\) for a simple proof.

Remark