Alternating mapping functions

We consider functions f from $n]:=\{1,2,\dots ,n\}$ into itself satisfying that the labels along the iteration orbit of each $i\in n]$ are forming an alternating sequence, i.e., $i<f(i)>f^2(i)<f^3(i)>?$ or $i>f(i)<f^2(i)>f^3(i)<?$ . We are able to solve the enumeration problem by stating exact and asymptotic formulæ for the number of such so-called alternating n-mapping functions. Furthermore we study the expected component structure of a randomly chosen alternating n-mapping by determining the probability that the underlying mapping graph is connected as well as the limiting distribution of the number of components. Moreover, the corresponding enumeration problem for weakly alternating n-mapping functions has also been solved.