\[ \DeclareMathOperator{\rel}{rel} \]

Let \(k\ge 3\) and \(\Pi_k\) be the set of all pairings on \(\{1,\ldots ,2k\}\). Suppose \(a,b,p\in \{1,\ldots ,2k\}\) are distinct, and let \(\pi\in \Pi_k\) be such that the deletion map \(\rho^p_{a,b}(\pi)\) is defined. Let \(\rel_{a,b}\) denote the relabeling map in the definition of \(\rho_{a,b}^p\). For brevity, we write \(\tilde{n}:=\rel_{a,b}(n)\). Then, we have the relation

\[ \rho_{a,b}^{\pi(p)}(\pi) = \tau_{\tilde{p},\tilde{\pi(p)}}(\rho_{a,b}^p(\pi)). \]

See also Link to heading

• deletion maps onto pairings
• transposition map onto pairings