Let \(k\ge 1\) and let \(\Pi_k\) denote the set of all pairings of \(\{1,\ldots ,2k\}\). For \(a,b\in \{1,\ldots ,2k\}\) with \(a\neq b\), we define the transposition map \(\tau_{a,b}:\Pi_k\to \Pi_k\) by
\[ \tau_{a,b}(\pi) = \begin{cases} \pi' & \text{if } b\neq \pi(a), \\ \pi & \text{if } b=\pi(a), \end{cases} \]where \(\pi'\) is obtained from \(\pi\) by replacing the pairs \((a,\pi(a))\) and \((b, \pi(b))\) with \((a,\pi(b))\) and \((b, \pi(a))\). Thus, \(\tau_{a,b}\) exchanges the partners of \(a\) and \(b\) in \(\pi\), and acts trivially if \(a\) and \(b\) are paired in \(\pi\).
We call \(\tau_{a,a+1}\) an adjacent transposition on \(\Pi_k\) and we write \(\tau_a\) instead of \(\tau_{a,a+1}\).
A transposition \(\tau_{a,b}\) is neutral with respect to a pairing \(\pi\in \Pi_k\) if \(b=\pi(a)\), it is positive if it is not neutral and if \(a< b\) is equivalent to \(\pi(a)< \pi(b)\), and it is negative if it is not neutral and \(a< b\) is equivalent to \(\pi(a)>\pi(b)\).
- The following relations are valid for \(a,b, c\in \{1,\ldots ,2k\}\), which are pairwise distinct,
- From the definition follows, that if \(\tau_{a,b}\) is positive with respect to a pairing \(\pi\in \Pi_k\), then \(\tau_{a,b}\) is negative with respect to \(\tau_{a,b}(\pi)\), and vice versa.
- Positive adjacent transpositions increase the pairing complexity by one, while negative adjacent transpositions reduce it by one (see (0x68980e37) ).
- For each pairing, except the maximal one, there is a positive adjacent transposition, and for each pairing, except the minimal one, there is a negative adjacent transposition (see (0x68979885) ).
- If we start from the minimal pairing, we can obtain every pairing by applying a finite sequence of positive adjacent transpositions, and we can reach every pairing by applying a finite sequence of negative adjacent transpositions if we start from the maximal pairing (see (0x687552d6) ).