Enumeration of order preserving maps (Q1205152)

If \(X\) is an ordered set, then \(\text{End}(X)\) denotes the set of all endomorphisms of \(X\) (i.e. order-preserving maps of \(X\) to \(X\)). Let \(e(X)=|\text{End}(X)|\). For a fixed positive integer \(n\), let \(e_ n=\min(e(X); | X|=n)\). The authors prove that for \(n\geq 3\), \(e_ n\geq (2^{2/3})^ n\) (and in the case of length one, there are at least \(2^ n\) endomorphisms). Furthermore, asymptotic estimates for the numbers of endomorphisms of crowns and fences are given. The exponential character of an infinite class of finite ordered sets is defined, and it is computed for several classes of ordered sets inspired by crowns and fences. The authors use interesting (among others, probabilistic) methods.