MathDB
CMI entrance 19#1

Source:

October 31, 2019
functionCMIcombinatoricscounting

Problem Statement

For a natural number nn denote by Map (n)( n ) the set of all functions f:{1,2,3,⋯ ,n}→{1,2,3,⋯ ,n}.f : \{ 1 , 2 , 3 , \cdots , n \} \rightarrow \{ 1 , 2 , 3 , \cdots , n \} . For f,g∈ f , g \in Map(n),f∘g( n ) , f \circ g denotes the function in Map (n)( n ) that sends x→f(g(x)).x \rightarrow f ( g ( x ) ) . \\ \\ (a)(a) Let f∈ f \in Map (n).( n ) . If for all x∈{1,2,3,⋯ ,n}f(x)≠x,x \in \{ 1 , 2 , 3 , \cdots , n \} f ( x ) \neq x , show that f∘f≠f f \circ f \neq f \\(b)(b) Count the number of functions f∈ f \in Map (n)( n ) such that f∘f=f f \circ f = f
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