Let \(k\in \mathbb{N}\) and \(\Pi_k\) be the set of pairings of \(\{1,\ldots ,2k\}\). For every \(\pi\in \Pi_k\) pairing complexity satisfies
\[ \mathcal{C}(\pi)\in [0,4k^2-2k], \]where only the minimal and the maximal pairing attain the border cases.
Since \(\pi_{k,\min }\) does not have any crossed or nested pairs, we have
\[ \mathcal{C}(\pi_{k,\min })=0. \]In \(\pi_{k,\max }\) all pairs are nested. Since there are \(\binom{2k}{2}\) options of choosing two pairs, the pairing complexity of \(\pi_{k,\max }\) is
\[ \mathcal{C}(\pi_{k,\max })=2 \binom{2k}{2}=2k(2k-1) = 4k^2-2k. \]Since for every pairing there is a negative adjacent transposition , except for the minimal pairing, and negative adjacent transpositions reduce pairing complexity by one , the pairing complexity of every pairing is positive, except the minimal one. Similarly,
\[ \mathcal{C}(\pi)< 4k^2-2k \]for every \(\pi\in \Pi_k\) with \(\pi\neq \pi_{k,\max }\).