Falling factorial, $(x)_n$, is the product of biggest $n$ terms in factorial, $(x)_n = x(x-1)(x-2)\cdot \ldots \cdot (x-n+1)$. Or the number of ways to color the set of $n$ objects into different colors if you have $x$ possible colors.
Using formula for probabilities of Poisson random variable $X$ one can find that $E((X)_n)=\lambda^n$. Derivation with $\lambda=1$. One may also use probability generating function. But I guess the answer is too beautiful to obtain it by boring summation :)
Starting from three axioms one may find $E(X)=\lambda$ without calculating probabilities. Just divide the interval $[0;1]$ into $n$ parts and make $n$ goes to infinity.
I wonder whether it is possible to find all the expected values $E((X)_n)$ directly from the defining axioms of Poisson process (without derivation of probabilities)???
not a homework — just curiosity :)