Let \(k\ge 1\), \(\Pi_k\) be the set of pairings on \(\{1,\ldots ,2k\}\).
The map \(\mathcal{P}_k\colon \Pi_k\to \mathbb{Z}[t]\) described in (0x68624c9f) is well-defined.
Warning
This not proven yet. However, this must be true from motivation.
Note
ChatGPT mentioned, that is suffices to check that \(\mathcal{P}_k\) respected the relations of the symmetric group which acts on \(\Pi_k\).