Let $(W, S)$ be a Coxeter system, and let $T = \bigcup\_{w \in W, s \in S} wsw^{-1}$. Associated to every element $t \in T$ is a unique positive root $\alpha\_t \in \Phi^{+}$ considered as a vector in the standard geometric representation $V$ of $W$. A total order on $T$ is a **reflection order** if, whenever $\alpha\_{t\_1} < \alpha\_{t\_2}$, it follows that $\alpha\_{t\_1} < x \alpha\_{t\_1} + y \alpha\_{t\_2} < \alpha\_{t\_2}$ whenever the middle term is a positive root with $x > 0, y > 0$. (See, for example, [Bjorner and Brenti's](http://books.google.com/books?id=OsaeruFjCRsC&printsec=frontcover&dq=bjorner+and+brenti&source=bl&ots=mvBZNqSImQ&sig=wNGg-W9sBQa4547x8-g1CGmbJFM&hl=en&ei=De08TOjcEoeglAeek5DqCA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBcQ6AEwAA#v=onepage&q=bjorner%20and%20brenti&f=false) book.)


Fix a reflection order and let $[u, v]$ be a Bruhat interval. A maximal chain $u = w\_0 \to w\_1 \to ... \to w\_m = v$ in the Bruhat order is what I'll call monotonic if $w\_i w\_{i-1}^{-1} > w\_{i+1} w\_i^{-1}$ in the reflection order. 


There is a nonrecursive formula for the Kazhdan-Lusztig polynomials $P\_{u,v}(q)$ which implies that $P\_{u,v}(0)$ is equal to the number of monotonic maximal chains in $[u, v]$. This number is known by other means to be equal to $1$, so I know that there is a unique monotonic maximal chain; however, I can't prove this directly. So far all I've been able to do is use the greedy algorithm to prove that at least one monotonic maximal chain exists.


Does anyone have a direct proof of this fact? 


**Edit:** No progress, but now I have a more general conjecture which I no longer know by other means is true. Fix a sequence $a\_1, ... a\_m$ of odd positive integers such that $\sum\_i a\_i = \ell(v) - \ell(u)$. Then there exists at most one monotonic chain (not necessarily maximal) such that $w\_i w\_{i-1}^{-1} \in T$ and such that $\ell(w\_{i-1}) - \ell(w\_i) = a\_i$. 



Monotonic maximal chains in a Coxeter group
Hi, you may want to try to peruse these two papers:


Dyer, M. J. Hecke algebras and shellings of Bruhat intervals. Compositio Math. 89 (1993), no. 1, 91--115. 


Dyer, M. J. Hecke algebras and shellings of Bruhat intervals. II. Twisted Bruhat orders. Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989), 141--165, Contemp. Math., 139, Amer. Math. Soc., Providence, RI, 1992.


Dyer proves that Bruhat intervals are EL-shellable, and this gives the answer to your first question. As to your more general question, there may be answers to those as well (indirectly of course).


