Towers of powers and Bruhat order (Q1199774)

In this short paper, the authors provide a counterexample \((23415<_ T 32145, 23415\|_ B 32145)\) showing that for \(n\geq 5\) the dual Bruhat order and Brunson's tower order for iterated exponentials are not the same (as they are for \(n\leq 4)\). This example allows them to settle several other conjectures concerning tower orders as well as to point out certain errors in the literature. The example makes use of an interesting theorem on iterated exponentials that should be useful in other contexts also.