On the lengths of zigzags in thin complexes

We consider zigzags in thin complexes. The main result states that the sum of the lengths of all zigzags in an $n$-complexe is equal to the sum of the lengths of all zigzags in all $(n-1)$-faces of this complex, and this sum also is the twice of the sum of the lengths of all zigzags in all $(n-2)$-faces. For simplicial and cubical $n$-complexes, the sum depends on the rank $n$ and the number of $(n-1)$-faces only. We also describe the sum of the lengths of all generalized zigzags, it depends on the rank and the number of flags. As an application, we find the number of zigzags in Coxeter complexes.