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We can identify each pairing in \(\Pi_k\) with a fixed-point-free involution of \(S_{2k}\). To be more precise, there is a map \(\pi'\colon \{1,\ldots ,2k\}\to \{1,\ldots ,2k\}\) for each \(\pi\in \Pi_k\) satisfying

1. \(\pi'(i)\neq i\) for all \(i\in \{1,\ldots ,2k\}\) and
2. \(\pi'(\pi'(i))=i\) for all \(i\in \{1,\ldots ,2k\}\).

A pair in \(\pi\) is then \(\{i,\pi'(i)\}\). This also motivates the convention to write \(\pi(i)\) for the partner of \(i\). Using cycling notation, we can write

instead of \(\pi=\{\{1,\pi(1)\},\{2,\pi(2)\},\ldots \}\).