Let \( m_0=n \). For each \( i\geq 0 \), choose a number \( x \) in \( \{1,\dots, m_i\} \) uniformly at random and let \( m_{i+1}= m_i – x\). This gives a random vector \( \mathbf{m}=(m_1,m_2, \dots) \). For each \( 1\leq k\leq n\), let \( X_k \) be the number of occurrences of \( k \) in the vector \( \mathbf{m} \).

For each \(1\leq k\leq n\), let \(Y_k\) be the number of cycles of length \(k\) in a permutation of \( \{1,\dots, n\} \) chosen uniformly at random. Prove that \( X_k \) and \(Y_k\) have the same distribution.

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