This is a problem I made unless it already existed. But I can't find the answer. Could someone help?
Question 1.
$A=\{1,2,3,\dots,n\},\quad B=\{f|f:A\mapsto A,f\space is\space a\space bijection\},\\ F(m,n)=\text{number of functions}\in B\text{ that can be expressed as } f^m\space where\space f\in B\\ \text{Find generalized expression for }F(m,n)$
Question 2.
$A=\{1,2,3,\dots,n\},\quad C=\{g|g:A\mapsto A,f\space is\space any\space function\},\\ G(m,n)=\text{number of functions}\in C\text{ that can be expressed as } g^m\space where\space g\in C\\ \text{Find generalized expression for }G(m,n)$
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As somebody voted to close, let me add some explanation on why I am asking this question. I tried to calculate $F(5,2)$ and couldn't find a easy way to get the answer. So I made a program to calculate for different cases and I found that the behavior is not as I expected. See some numbers below:
$$F(1,5)=120, F(2,5)=60, F(3,5)=80, F(4,5)=45, F(5,5)=96\\ F(6,5)=40, F(7,5)=120, F(8,5)=45, F(9,5)=80, F(10,5)=46$$ $$\\$$ $$G(1,5)=3125, G(2,5)=1075, G(3,5)=985, G(4,5)=580, G(5,5)=1281\\ G(6,5)=295, G(7,5)=1305, G(8,5)=580, G(9,5)=925, G(10,5)=631\\ \dots\\G(96,5)=220, G(97,5)=1305, G(98,5)=655, G(99,5)=925, G(100,5)=556$$
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I don't see any clear trend or pattern here. Even $G(m,5)$ (that I thought would monotonically decrease) doesn't keep decreasing. So I thought it may be interesting to see if there's something that can explain this behavior clearly. Thanks.