In this thesis, we extend the work of fixed points on the permutations of [n] in two directions: firstly, we discuss the fixed points problems of hyperoctahedral groups Bn; secondly, elements in Bn can be thought the letters
are painted by two colors, it can be generalized with r colors. Moreover, we discuss the fixed point problems in the subsets alternating permutations of Bn and strictly decreasing permutations with r colors. After removing all fixed points and standardizing the remaining letters, we focus on colored permutations with 0 or 1 fixed point. We obtain combinatorial correspondence between derangements and elements with exactly one fixed point together with their recursions and generating functions.

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