Let \(k\ge 2\), \(\Pi_k\) be the set of pairings of \(\{1,\ldots ,2k\}\) and \(\mathcal{P}_k\colon \Pi_k\to \mathbb{Z}[t]\) the map described in (0x68624c9f) .
If \(\{1,2\}\in \pi\) for some \(\pi\in \Pi_k\) then
\[ \mathcal{P}_k(\pi)=t\mathcal{P}_{k-1}(\widetilde{\pi}) \]where \(\widetilde{\pi}\in \Pi_{k-1}\) is the pairing corresponding to \(\pi\setminus \{\{1,2\}\}\).
Proof
Use a proof by induction.
Remarks
By induction follows if \(\{\{1,2\}, \{3,4\}, \ldots , \{2m-1, 2m\}\}\in \pi\in \Pi_k\) for some \(m< k\), then
\[ \mathcal{P}_k(\pi) = t^m \,\mathcal{P}_{k-m}(\widetilde{\pi}) \]where \(\widetilde{\pi}\in \Pi_{k-m}\) is the pairing corresponding to \(\pi\setminus \{\{1,2\}, \{3,4\}, \ldots , \{2m-1, 2m\}\}\).