  This note focuses on the properties of two blocks of elements of the
probability mass function (pmf) of the Poisson distribution of order $k\ge2$.
The first block is the elements for $n\in[1,k]$ and the second block is the
elements for $n\in[k+1,2k]$. It is proved that elements in the first block form
an ``absolutely monotonic sequence'' by which is meant that all the finite
differences of the sequence are positive. Next, the properties of the elements
in the second block are analyzed. It is shown that for sufficiently small
$\lambda>0$, the sequence of elements for $n\in[k+1,2k]$ is strictly decreasing
and also concave. The purpose of the analysis is to help determine a supremum
value for $\lambda$, such that the pmf of the Poisson distribution of order
$k\ge2$ decreases strictly for all $n \ge k$. A conjectured criterion for the
supremum is given. Numerical calculations indicate it is the optimal bound,
i.e.~the supremum. In addition, a simple expression is proposed, based on
numerical calculation, which is sufficient (but not necessary) and is a good
approximation for the supremum.
