A note on association schemes with two \(\text{P}\)-polynomial structures of type III (Q1914005)

Let \({\mathcal X}= (\Gamma, {\mathbf R})\) be a P-polynomial association \(d\)-scheme. Eiichi Bannai and Etsuko Bannai proved that if \(\mathcal X\) is not a polygon and has another P-polynomial structure, then there are four types (I)--(IV) for P-polynomial orderings. In case (III) the P-polynomial ordering is: \(R_0, R_d, R_2, R_{d- 2},\dots, R_{d- 3}, R_3, R_{d- 1}, R_1\). Brouwer proved that in this case \(p^j_{1j}= 0\) for all \(j\) if \(d\geq 7\). Theorem 1.2. Let \({\mathcal X}= (\Gamma, {\mathbf R})\) be a P-polynomial association \(d\)-scheme with respect to the natural ordering. If \(R_0, R_d, R_2, R_{d- 2},\dots, R_{d- 3}, R_3, R_{d- 1}, R_1\) defines another P-polynomial structure, then \(d= 3\) or 4.