The simplest sequential description of the Ewens sampling formula is called the Chinese restaurant process. The first

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The simplest sequential description of the Ewens sampling formula is called the Chinese restaurant process. The first customer arrives and is seated at a table. After \(n\) customers have been seated, the next customer is seated alone with probability \(\alpha /(n+\alpha)\); otherwise, the newcomer selects one of the seated customers uniformly at random and sits at that table. Show that the configuration after \(n\) customers are seated is given by \(P_{n, \alpha}\). Hence deduce that customers seven and nine are seated together with probability \(1 /(\alpha(\alpha+1))\).

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