  The Poisson distribution of order $k$ is a special case of a compound Poisson
distribution. For $k=1$ it is the standard Poisson distribution. Our main
result is a proof that for sufficiently small values of the rate parameter
$\lambda$, the probability mass function (pmf) decreases monotonically for all
$n\ge k$ (it is known that the pmf increases strictly for $1\le n \le k$, for
fixed $k\ge2$ and all $\lambda>0$). The second main result is a partial proof
that the difference (mean $-$ mode) does not exceed $k$. The term `partial
proof' signifies that the derivation is conditional on an assumption which,
although plausible and supported by numerical evidence, is as yet not proved.
This note also presents new inequalities, and sharper bounds for some published
inequalities, for the Poisson distribution of order $k$.
