KMO 2021 Problem 2

For positive integers $n, k, r$ , denote by $A(n, k, r)$ the number of integer tuples $(x_1, x_2, \ldots, x_k)$ satisfying the following conditions.

• $x_1 \ge x_2 \ge \cdots \ge x_k \ge 0$
• $x_1+x_2+ \cdots +x_k = n$
• $x_1-x_k \le r$

For all positive integers $m, s, t$ , prove that $$A(m, s, t)=A(m, t, s).$$