Canonical reduced words and signed descent length enumeration in Coxeter groups (Q6639848)

Let \(\mathfrak{S}_{n}\) be the symmetric group on \([n] =\{1,2, \ldots , n \}\). \textit{T. K. Petersen} and \textit{B. E. Tenner}, in [J. Comb. 6, No. 1--2, 145--178 (2015; Zbl 1317.20040)], present a statistic called depth for elements of \(\mathfrak{S}_{n}\), defined in terms of factorizations of the elements into products of reflections.\N\NIn the paper under review, the authors, using descents in canonical reduced words of elements in \(\mathfrak{S}_{n}\), give an involution \(f_{A}: \mathfrak{S}_{n} \rightarrow \mathfrak{S}_{n}\) that leads to a neat formula for the signed trivariate enumerator of drops, depth, exc in \(\mathfrak{S}_{n}\) (see paper for precise definitions). This gives a simple formula for the signed univariate drops enumerator in \(\mathfrak{S}_{n}\). Using similar techniques, the authors show analogous univariate results for type-B Coxeter groups and for the type-D Coxeter groups, they get analogous but inductive univariate results. Under the Foata-Zeilberger bijection \(\phi_{FZ}F\) which takes permutations to restricted Laguerre histories (see [\textit{D. Foata} and \textit{D. Zeilberger}, Stud. Appl. Math. 83, No. 1, 31--59 (1990; Zbl 0738.05001)]), they show that permutations \(\pi\) and \(f_{A}(\pi)\) map to the same Motzkin path, but have different history components. Using the Foata-Zeilberger bijection, they also get a continued fraction for the generating function enumerating the pair of statistics drops and MAD.