0x6880caad

Let \(k\ge 1\), \(\Pi_k\) be the set of pairings on \(\{1,\ldots ,2k\}\) and \(\mathcal{P}_k\colon \Pi_k\to \mathbb{Z}[t]\) the map described in (0x68624c9f) .

For every pairing \(\pi\in \Pi_k\), the polynomial \(\mathcal{P}_k(\pi)\) has only nonnegative coefficients.

We abbreviate this by writing

\[ \mathcal{P}_k(\pi)\ge 0. \]