Leetcode 3317. Find the Number of Possible Ways for an Event

Count the ways to assign n labeled performers to x distinguishable stages (some stages may be empty) and then give each non-empty stage a score from 1..y, with the total taken modulo 1e9+7. Equivalently, compute the sum over k (number of non-empty stages) of C(x,k)*S(n,k)*k!*y^k, so the main challenge is efficiently computing Stirling numbers and binomial coefficients under the modulus for n,x,y up to 1000.