Let \(k\in \mathbb{N}\) and \(\Pi_k\) be the set of pairings of \(\{1,\ldots ,2k\}\). We call \(\pi_{k,\min}=(1~~2)(3~~4)\cdots (2k-1~~2k)\) minimal pairing of \(\Pi_k\) and \(\pi_{k,\max } = (1~~2k)(2~~2k-1)\cdots\) maximal pairing of \(\Pi_k\).

Remarks
  • 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) ).
  • The maximal pairing \(\pi_{k,\max }\) has maximal complexity in \(\Pi_k\) and the minimal pairing \(\pi_{k,\min }\) has minimal complexity in \(\Pi_k\) (see (0x68984230) ). This is the reason why we call the pairings that way.