  Kostadinova and Minkova published an expression for the probability mass
function (pmf) of the Poisson distribution of order $k$, as a combinatorial sum
($\mathit{Pliska~Stud.~Math.~Bulgar.}\ {\bf 22},\ 117-128\ (2013)$). Inspired
by their elegant solution, this note presents an alternative combinatorial sum
for the pmf of the Poisson distribution of order $k$. The terms are partitioned
into blocks of length $k$ (as opposed to $k+1$ by Kostadinova and Minkova). The
new sum offers an advantage in the following sense. For $n\in[rk+1,(r+1)k]$,
the lowest power of $\lambda$ in the pmf is $\lambda^{r+1}$. Hence the lower
limit of summation can be increased, to avoid needlessly calculating terms
which cancel to identically zero.
