Let \(k\in \mathbb{N}\), and let \(\Pi_k\) denote the set of all pairings on \(\{1,\ldots ,2k\}\). Let \(\mathcal{P}_k\colon \Pi_k\to \mathbb{Z}[t]\) denote the map defined in (0x68624c9f) .
There are two numerical constants such that for every \(E>0\) we have
\[ \sup_{\pi\in \Pi_k} \lVert \mathcal{P}_k(\pi)\rVert_{L^\infty ([0,E])} \le C_1^kE^k + C_2^k (k!)^2. \]
Proof