  We suggest a new hardcore Poisson-type distribution for Young diagrams with
the row lengths from some finite list. A discrete variant of the time-ordered
Mat\'{e}rn II process in 1D is employed. This approach is related to that based
on the interlacing sequences due to Kerov and others, but we restrict the
number of rows. The basic lengths are assumed comparable with the total order
of the diagram in the quasi-classical limit, which results in new methods and
new formulas. An interesting application is to random walks where the steps are
at the points satisfying the classical Poisson distribution or our
truncatedone. In the simplest case, one obtains the distribution in terms of
Bessel I-functions, which provides some probabilistic interpretation of its
many properties. An immediate application of our truncated Poisson
distributions is to modeling reinfections in epidemics.
