Polynomial Map onto Pairings

\[ \DeclareMathOperator{\rel}{rel} \DeclareMathOperator{\ran}{ran} \]

Let \(k, d\in \mathbb{N}\) and let \(\Pi_k\) denote the set of all pairings of \(\{1,\ldots ,2k\}\). We recursively define a map

\[ \mathcal{P}_k\colon \Pi_k \to \mathbb{Z}[t] \]

as follows:

1. For the minimal pairing \(\pi_{k,\min }\) we set \[ \mathcal{P}_k(\pi_{k,\min })= t^k. \]
2. Let \(\pi\in \Pi_k\). For distinct \(a,b\in \{1,\ldots ,2k\}\), we introduce the notation \[ \Delta_{a,b}(\pi):=\mathcal{P}_k(\tau_{a,b}(\pi))-\mathcal{P}_k(\pi). \] Moreover, let \(\rel_a \) denote the relabeling map in the definition of the deletion map \(\rho_a^p\). For brevity, we write \(n':=\rel_a(n)\). Let \(a\in \{1,\ldots ,2k-1\}\). If the adjacent transposition \(\tau_a\) is not neutral with respect to \(\pi\), we define \[ \Delta_a(\pi) := \sum_{\substack{p< a, \\ p \neq \pi(a), \pi(a+1), \\ \pi(p)>a}} \Delta_{p', \pi(p)'}\bigl(\rho^p_a(\pi)\bigr) + \epsilon_a(\pi) \mathcal{P}_{k-1}\bigl(R_a(\pi)\bigr), \] where \(R_a\) is a Ricci delete on \(\Pi_k\), and \[ \epsilon_a(\pi) = \begin{cases} d & \text{if } \pi(a)< a < \pi(a+1), \\ -d & \text{if } \pi(a+1)< a < \pi(a), \\ 0 & \text{otherwise}. \end{cases} \] In the case that \(\tau_a\) is neutral with respect to \(\pi\), we set \(\Delta_a(\pi)=0\). Then, \[ \mathcal{P}_k\bigl(\tau_a(\pi)\bigr) = \mathcal{P}_k(\pi) + \Delta_a(\pi). \]