  We study the problem of the decay of initial data in the form of a unit step
for the Bogoyavlensky lattices. In contrast to the Gurevich--Pitaevskii problem
of the decay of initial discontinuity for the KdV equation, it turns out to be
exactly solvable, since the dynamics is linearizable due to termination on the
half-line. The answer is written in terms of generalized hypergeometric
functions, which serve as exponential generating functions for generalized
Catalan numbers. This can be proved by the fact that the generalized Hankel
determinants for these numbers are equal to 1, which is a well-known result in
combinatorics. Another method is based on a non-autonomous symmetry reduction
consistent with the dynamics. It reduces the lattice equation to a
finite-dimensional system and makes it possible to solve the problem for a more
general finite-parameter family of initial data.
