论文标题
签名和签名的置换组的Worpitzky身份
The Worpitzky identity for the groups of signed and even-signed permutations
论文作者
论文摘要
The well-known Worpitzky identity provides a connection between two bases of $\mathbb{Q}[x]$: The standard basis $(x+1)^n$ and the binomial basis ${{x+n-i} \choose {n}}$, where the Eulerian numbers for the Coxeter group of type $A$ (the symmetric group) serve as the entries of the transformation matrix. Brenti已使用生成功能技术将这种身份概括为$ b $ and $ b $和$ d $（分别签名且均匀签名的排列组）的Coxeter组。 由Foata-Schützenberger和Rawlings在对称组中的Worpitzky身份证明的动机，我们提供了这种身份的组合证明以及其在Coxeter类型$ b $和$ $ b $和$ d $中的$ q- $类似物。
The well-known Worpitzky identity provides a connection between two bases of $\mathbb{Q}[x]$: The standard basis $(x+1)^n$ and the binomial basis ${{x+n-i} \choose {n}}$, where the Eulerian numbers for the Coxeter group of type $A$ (the symmetric group) serve as the entries of the transformation matrix. Brenti has generalized this identity to the Coxeter groups of types $B$ and $D$ (signed and even-signed permutations groups, respectively) using generating function techniques. Motivated by Foata-Schützenberger and Rawlings' proof for the Worpitzky identity in the symmetric group, we provide combinatorial proofs of this identity and for their $q-$analogues in the Coxeter groups of types $B$ and $D$.
