0x68980e37

Suppose \(\pi\in \Pi_k\) and \(\tau_a\) is a positive adjacent transposition with respect to \(\pi\) for some \(a\in \{1,\ldots ,2k-1\}\), i.e. \(\pi(a)< \pi(a+1)\). Note, that it is not clear what relationship \(\pi(a)\) and \(\pi(a+1)\) have with \(a\) and \(a+1\). For simplicity, we write \(\widetilde{\pi}=\tau_a(\pi)\).

It suffices to check how the relation of the pairs \(\{a,\pi(a)\}\) and \(\{a+1,\pi(a+1)\}\) changes after applying \(\tau_a\), because, if \(\{a,\pi(a)\}\) is crossed (or nested) with some other pair in \(\pi\), this pair will be crossed (or nested) with \(\{a+1,\widetilde{\pi}(a+1)\}\) in \(\widetilde{\pi}\), and vice versa. All other crossed and nested pairs are not affected by \(\tau_a\).

As mentioned above, there are several options how \(\pi(a)\), \(\pi(a+1)\), \(a\) and \(a+1\) could be arranged. If

\[ \pi(a)< \pi(a+1)< a< a+1, \]

then \(\{a,\pi(a)\}\) and \(\{a+1,\pi(a+1)\}\) are crossed in \(\pi\). After applying \(\tau_a\), we get

\[ \widetilde{\pi}(a+1) < \widetilde{\pi}(a) < a < a+1, \]

and the pairs \(\{a,\widetilde{\pi}(a)\}\) and \(\{a+1, \widetilde{\pi}(a+1)\}\) are nested in \(\widetilde{\pi}\). Therefore, \(\mathcal{C}(\widetilde{\pi})=\mathcal{C}(\pi)+1\).

Similarly, if

\[ \pi(a)< a< a+1< \pi(a+1), \]

then \(\{a,\pi(a)\}\) and \(\{a+1,\pi(a+1)\}\) become crossed in \(\widetilde{\pi}\) and again \(\mathcal{C}(\widetilde{\pi})=\mathcal{C}(\pi)+1\). If

\[ a< a+1< \pi(a)< \pi(a+1), \]

then the crossed pairs \(\{a,\pi(a)\}\) and \(\{a+1,\pi(a+1)\}\) become nested in \(\widetilde{\pi}\). This also yields \(\mathcal{C}(\widetilde{\pi})=\mathcal{C}(\pi)+1\).

If \(\tau_a\) is a negative adjacent transposition with respect to \(\pi\), we use the fact that \(\tau_a\) is a positive adjacent transposition with respect to \(\tau_a(\pi)\). Then,

\[ \mathcal{C}(\tau_a\circ \tau_a(\pi)) = \mathcal{C}(\tau_a(\pi))+1. \]

Since \(\tau_a\circ \tau_a(\pi)=\pi\) , we obtain

\[ \mathcal{C}(\tau_a(\pi))=\mathcal{C}(\pi)-1. \]