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5

Part of 2008 Pre-Preparation Course Examination

Problems(1)

Cycles and Permutations

Source: Iran PPCE 2008

12/29/2008
A permutation π \pi is selected randomly through all n n-permutations. a) if C_a(\pi)\equal{}\mbox{the number of cycles of length }a\mbox{ in }\pi then prove that E(C_a(\pi))\equal{}\frac1a b) Prove that if {a1,a2,…,ak}⊂{1,2,…,n} \{a_1,a_2,\dots,a_k\}\subset\{1,2,\dots,n\} the probability that π \pi does not have any cycle with lengths a1,…,ak a_1,\dots,a_k is at most \frac1{\sum_{i\equal{}1}^ka_i}
probabilityfactorialexpected valueprobability and stats