  Relations between the mean values of distributions of flipped spins on
periodic Heisenberg XX chain and some aspects of enumerative combinatorics are
discussed. The Bethe vectors, which are the state-vectors of the model, are
considered both as on- and off-shell. It is this approach that makes it
possible to represent and to study the correlation functions in the form of
non-intersecting nests of lattice walks and related plane partitions. We
distinguish between two types of walkers, namely lock step models and random
turns. Of particular interest is the connection of random turns walks and a
circulant matrix. The determinantal representation for the norm-trace
generating function of plane partitions with fixed height of diagonal parts is
obtained as the expectation of the generating exponential over off-shell
N-particle Bethe states. The asymptotics of the dynamical mean value of the
generating exponential is calculated in the double scaling limit provided that
the evolution parameter is large. It is shown that the amplitudes of the
leading asymptotics depend on the number of diagonally constrained plane
partitions.
