Let \(k\ge 2\) and \(\Pi_k\) be the set of all pairings of \(\{1,\ldots ,2k\}\).
For distinct \(a,b\in \{1,\ldots ,2k\}\), we define the Ricci delete
\[ R_{a,b}\colon \Pi_k \to \Pi_{k-1}. \]If \(\pi\in \Pi_k\) and \(a\) and \(b\) are not paired in \(\pi\), then
\[ R_{a,b}(\pi) = \pi', \]where \(\pi'\) is obtained from \(\pi\) by
- removing the pairs \((a,\pi(a))\) and \((b,\pi(b))\) and replacing them with the pair \((\pi(a), \pi(b))\);
- relabeling the elements of \(\{1,\ldots ,2k\}\setminus \{a,b\}\) by elements of \(\{1,\ldots ,2k-2\}\) in increasing order.
Explicitly, if \(n\in \{1,\ldots ,2k\}\setminus \{a,b\}\), we define
\[ \rel_{a,b}(n):=\begin{cases} n & \text{if } n< \min \{a,b\},\\ n-1 & \text{if } \min \{a,b\}< n< \max \{a,b\},\\ n-2 & \text{if } n > \max \{a,b\}, \end{cases} \]and obtain \(\pi'\) by replacing each entry \(n\) with \(\rel_{a,b}(n)\). If \(a\) and \(b\) are already paired in \(\pi\), \(R_{a,b}\) is not defined.
Suppose \(\pi=(1~~2)(3~~4)(5~~6)\in \Pi_3\). To determine \(R_{2,5}(\pi)\)), we first replace \((1,2)\) and \((5,6)\) by \((1,6)\) which gives us \((3~~4)(1~~6)\). Relabeling yields
\[ R_{2,5}(\pi)=(1~~4)(2~~3) \in \Pi_2. \]For distinct \(a,b,p\in \{1,\ldots ,2k\}\) and \(k\ge 3\), we define the deletion map
\[ \rho_{a,b}^p\colon \Pi_k\to \Pi_{k-1}. \]If \(\pi\in \Pi_k\) and none of the pairs \((a,b)\), \((a,p)\) and \((b,p)\) belongs to \(\pi\), then
\[ \rho_{a,b}^p(\pi)=\pi', \]where \(\pi'\) is obtained from \(\pi\) by
- removing the pairs \((a,\pi(a))\), \((b,\pi(b))\) and \((p,\pi(p))\) and replacing them with the pairs \((p,\pi(a))\) and \((\pi(p),\pi(b))\);
- relabeling the elements of \(\{1,\ldots ,2k\}\setminus \{a,b\}\) with \(\{1,\ldots ,2k-2\}\) in increasing order by replacing each entry \(n\) with \(\rel_{a,b}(n)\).
If \((a,b)\), \((a,p)\) or \((b,p)\) are already in \(\pi\), \(\rho^p_{a,b}\) is not defined.
Suppose \(\pi=(1~~2)(3~~4)(5~~6)\in \Pi_3\). To determine \(\rho^3_{2,5}(\pi)\)), we first replace \((1,2)\), \((3,4)\) and \((5,6)\) by \((1,3)\) and \((4,5)\) which gives us \((1~~3)(4~~6)\). Relabeling yields
\[ \rho_{2,5}^3(\pi) =(1~~2)(3~~4)\in \Pi_2. \]For brevity, we write \(R_a\) instead of \(R_{a,a+1}\), \(\rel_a\) instead of \(\rel_{a,a+1}\) and \(\rho_a^p\) instead of \(\rho_{a,a+1}^p\).
- For \(\pi\in \Pi_k\), we have \[ \rho_{a,b}^{\pi(p)}(\pi) = \tau_{\tilde{p},\tilde{\pi(p)}}(\rho_{a,b}^p(\pi)), \] (see (0x6881dd71) ).