论文标题
经典Weyl组的签名交替运行
Signed Alternating-runs enumeration in Classical Weyl Groups
论文作者
论文摘要
交替运行多项式列举对称组中的交替运行。对于排列数，有三个公式，$ r_ {n，k} $ in $ \ mathfrak {s} _n $带有$ k $交替运行，但所有这些都很复杂。我们表明，当考虑到符合符号的列举时，就会得到一个{\ it nyeat公式}。结果，当它将交替运行的多项式在交替的组$ \ mathcal {a} _n $中划分为$（1+t）$的指数时，我们几乎可以得到$（1+t）$的结果。其他应用程序包括$ \ MATHCAL {a} _n $中交替排列的矩类型身份。对于B型和D型Coxeter组获得了相似的结果。
The alternating-runs polynomial enumerates alternating runs in the symmetric group. There are three formulae for the number of permutations, $R_{n,k}$ in $\mathfrak{S}_n$ with $k$ alternating runs, but all of them are complicated. We show that when enumerated with sign taken into account, one gets a {\it neat formula}. As a consequence, we get a near refinement of a result of Wilf on the exponent of $(1+t)$ when it divides the alternating-runs polynomial in the alternating group $\mathcal{A}_n$. Other applications include a moment-type identity and enumeration of alternating permutations in $\mathcal{A}_n$. Similar results are obtained for the type B and type D Coxeter groups.
