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The Poisson distribution of order $k$ is a special case of a compound Poissondistribution. For $k=1$ it is the standard Poisson distribution. Our focus inthis note is for $k\ge2$. For sufficiently small values of the rate parameter$\lambda$, both the median and mode equal zero. The median is zero if and onlyif $\lambda \le (\ln2)/k$. The supremum value of $\lambda$ for the mode to bezero is known only for small values of $k$. This note presents results for the"first double mode" by which is meant the first occasion (smallest value of$\lambda$) the distribution is bimodal, with modes at $0$ and $m>0$. Next, analmost complete answer is supplied to the question "which positive integerscannot be modes of the Poisson distribution of order $k$?" The term "almostcomplete" signifies that some parts of the answer are conjectures based onnumerical searches. However, if the conjectures are proved to be correct, thesolution presented in this note is complete: all parameter values are covered.

First double mode of the Poisson distribution of order $k$