=+2. For a random permutation 1,...,n of {1,...,n}, let X = 1{=} be the indicator of a

=+2. For a random permutation σ1,...,σn of {1,...,n}, let Xα = 1{σα=α}

be the indicator of a match at position α. Show that the total number of matches S = n

α=1 Xα satisfies the coupling bound

πS − πZTV ≤ 2(1 − e−1)

n , where Z follows a Poisson distribution with mean 1.

14.5 Problems 369