We denote the set of all pairings of \(\{1,\ldots ,2k\}\) by \(\Pi_k\). An element of \(\pi\in \Pi_k\) is a pair in \(\pi\), and the two elements of a pair are called partners. For \(a\in \{1,\ldots ,2k\}\), the partner of \(a\) in \(\pi\) is denoted by \(\pi(a)\).
We identify each pairing with a fixed-point-free involution , and represent it using cycle notation.
Example
The pairing
\[ \pi=(1~~2)(3~~4)\cdots (2k-1~~2k) \]belongs to \(\Pi_k\).
Remarks
- Every pairing is a partition of \(\{1,\ldots ,2k\}\).
- \(\Pi_k\) has \((2k-1)!! = \prod_{i=1}^{k} (2i-1)\) elements.
- We can define a group action of \(S_{2k}\) on \(\Pi_k\) by \[ \phi_\sigma(\pi)= \sigma \pi \sigma^{-1}. \]
- One way to visualize a pairing is to connect the paired numbers by a line. The following figure gives an example.
Visualization of the pairing \((1~~6)(2~~4)(3~~5)\)