By direct calculation, show that the Ewens distributions satisfy the following conditions:

Show that \(P_{4, \alpha}\) is the marginal distribution of \(P_{5, \alpha}\) when the element 5 is removed from the set [5]. Hence calculate the conditional distribution \(P_{5, \alpha}(x \mid B)\) for \(x \in \mathcal{P}_{5}\) and \(B=1|3| 24\) and \(B=13 \mid 24\).

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P2.a (12) P3.a ({123, 12|3}) P2,a (1 2) P3,a ({13|2, 1|23, 1|2|3}) = P3, a (123) P4,a ({1234, 123|4}) P3,a (1123)= P4, a ({14|23, 1|234, 1|23|4}) P3,a (1|2|3) P4, ({14|2|3, 1|24|3, 1|2|34, 1|2|3|4}). = = =