Enumerative combinatorics
Handbook of combinatorics (vol. 2)
On discrete Morse functions and combinatorial decompositions
Discrete Mathematics
Journal of Algebraic Combinatorics: An International Journal
Permutation statistics on involutions
European Journal of Combinatorics
Pattern avoidance and Boolean elements in the Bruhat order on involutions
Journal of Algebraic Combinatorics: An International Journal
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Given a Coxeter system (W,S) equipped with an involutive automorphism 驴, the set of twisted identities is $$\iota (\theta )=\{\theta(w^{-1})w\mid w\in W\}.$$ We point out how 驴(驴) shows up in several contexts and prove that if there is no s驴S such that s 驴(s) is of odd order greater than 1, then the Bruhat order on 驴(驴) is a graded poset with rank function 驴 given by halving the Coxeter length. Under the same condition, it is shown that the order complexes of the open intervals either are PL spheres or 驴-acyclic. In the general case, contractibility is shown for certain classes of intervals. Furthermore, we demonstrate that sometimes these posets are not graded. For the Poincaré series of 驴(驴), i.e. its generating function with respect to 驴, a factorisation phenomenon is discussed.