Why for is even?
There exist at least two ways of convincing us about this fact:
1) Analyze such difference with the aid of the Newton's binomial in each case for combinations of parity: a) x is even and any y, b) x is odd and y is even, and c) Both odd.
2) To observe that such fact necessarily is true as direct consequence from the definition of "factorial". As follows:
By separating the first term () term:
It is straightforward to note due the presence of that the right-hand side is zero mod 2, always that x,y>1.
[[For reference: Wikipedia contributors. Stirling numbers of the second kind. Wikipedia, The Free Encyclopedia.]]