Let \(k, d\in N\), \(\Pi_k\) be the set of pairings on \(\{1,\ldots ,2k\}\) and \(\pi\in \Pi_k\). Suppose \(\mathcal{P}_k\) is the polynomial map described in (0x68624c9f)
Then the polynomial \(\mathcal{P}_k(\pi)\) has the following representation
\[ \mathcal{P}_k(\pi) = t^k + a_{k-1}t^{k-1} + \cdots + a_1t^1 \]with
\begin{align*} a_{k-1}&= b_{k-1} d,\\ a_{k-2}&= b_{k-2,2}d^2 + b_{k-2,1}d , \\ &\quad\vdots\\ a_1 &= b_{1,k-1} d^{k-1} + \cdots + b_{1,2}d^2 + b_{1,1}d. \end{align*}
Remarks
- \(b_{k-1}\) is the Ricci number, i.e. the number of positive Ricci moves from the starting pairing \(\pi_{k,0}\) to \(\pi\)(see (0x68754dc6) ).
- \(b_{k-2, 2}\) is the sum of Ricci numbers of all pairings obtained by Ricci deletions we use to obtain \(\pi\) starting from \(\pi_{k,0}\) and \(b_{k-2,1}\) corresponds to the Ricci number differences of the Ricci differences along the way from \(\pi_{k,0}\) to \(\pi\).