  A generalization of the Bethe ansatz equations is studied, where a scalar
two-particle S-matrix has several zeroes and poles in the complex plane, as
opposed to the ordinary single pole/zero case. For the repulsive case (no
complex roots), the main result is the enumeration of all distinct solutions to
the Bethe equations in terms of the Fuss-Catalan numbers. Two new combinatorial
interpretations of the Fuss-Catalan and related numbers are obtained. On the
one hand, they count regular orbits of the permutation group in certain factor
modules over Z^M, and on the other hand, they count integer points in certain
M-dimensional polytopes.
