Let \(k, d\in N\), \(\Pi_k\) be the set of pairings of \(\{1,\ldots ,2k\}\) and \(\pi\in \Pi_k\). Suppose \(\mathcal{P}_k\) is the polynomial map described in (0x68624c9f)
The coefficient of the monomial \(t^{k-1}\) of \(\mathcal{P}_k^d(\pi)\) is \( r_{\pi}d\), where \(r_{\pi}\) denotes the number of positive Ricci moves needed from the start partition \(\{\{1,2\},\{3,4\},\ldots ,\{2k-1,2k\}\}\) to \(\pi\).
We call \(r_\pi\) the Ricci number of \(\pi\).
Proof
Since all values of \(\mathcal{P}_{k-1}^d\) are monic, the difference in the sum are polynomials of order \(k-2\). Therefore, the only relavent terms are the ones produced by the reduction map \(R_a\) in the definition. Since we can obtain every pairing by adjacent transpositions of ascents (see (0x687552d6)
), \(\epsilon\) is either 0 or 1. Thus, the coefficient of the monomial \(t^{k-1}\) of \(\mathcal{P}_k^d(\pi)\) corresponds to \(r_\pi d\).
Remark
The Ricci number \(r_\pi\) corresponds to the number crossed and nested pairs \(\pi\), because a Ricci move generates a crossed pair and adjacent transpositions of ascents only turn crossed pairs in a nested ones.
See also Link to heading
- Coefficients Overview