On the lengths of zigzags in thin complexes
Abstract
We consider zigzags in thin complexes. The main result states that the sum of the lengths of all zigzags in an -complexe is equal to the sum of the lengths of all zigzags in all -faces of this complex, and this sum also is the twice of the sum of the lengths of all zigzags in all -faces. For simplicial and cubical -complexes, the sum depends on the rank and the number of -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.
Keywords
Cite
@article{arxiv.1602.05401,
title = {On the lengths of zigzags in thin complexes},
author = {Michel Deza and Mark Pankov},
journal= {arXiv preprint arXiv:1602.05401},
year = {2016}
}
Comments
The main result of the paper (Theorem 1) is a very simple consequence of the following trivial observation: the sum of the lengths of all zigzag is equal to the number of flags. So, it cannot be considered as a contribution