We give a classification of the regular soliton solutions of the KP hierarchy, referred to as the KP solitons, under the Gel’fand-Dickey \(\ell \) -reductions in terms of the permutation of the symmetric group. As an example, we show that the regular soliton solutions of the (good) Boussinesq equation as the 3-reduction can have at most one resonant soliton in addition to two sets of solitons propagating in opposite directions. We also give a systematic construction of these soliton solutions for the \(\ell \) -reductions using the vertex operators. In particular, we show that the non-crossing permutation gives the regularity condition for the soliton solutions.
