Structurally equivalent tuples in the equiprobable polynomial scheme

Let $X_1,\dots,X_n$ be a sequence of independent random variables with the uniform distribution on the set $\{1,\dots,N\}$. We describe limit discrete distributions of the number of $k$-element sets consisting of structurally equivalent $s$-tuples for $N,n,s\to\infty$, $sN^{-1}\to\alpha\in(0,1)$, $n(N)_sN^{-s}\to\lambda\in(0,\infty)$ and arbitrary $k\geqslant2$. The proofs are based on the Chen–Stein method.