论文标题
$ Q $ - 基于下降的均等的B类和D欧拉的多项式
$q$-enumeration of type B and D Eulerian polynomials based on parity of descents
论文作者
论文摘要
Carlitz和Scoville在1973年被认为是一个四个变量多项式，该多项式列举了$ \ Mathfrak {s} _n $在其下降和上升的奇偶校验中。在最近的工作中，Pan和Zeng证明了Carlitz-Scoville的生成功能的$ Q $ Analogue通过列举上述四个统计数据以及倒置编号来列举排列。此外，他们还通过列举了签名的置换，以相对于下降和上升的奇偶校验证明了B型类似物。在这项工作中，我们通过在上述四个统计信息和B类反转号码中列举了PAN和Zeng类型B结果的$ Q $ -Analogue。我们还获得了B型双变量交流多项式的生成功能的$ Q $ - 动态。我们在D型Coxeter组中也考虑了类似的五变量多项式，并给予其EGF。 Dype组的交替下降以前也由REMMEL定义，但我们的定义略有不同。作为我们证明的副产品，我们获得了Bivariate $ Q $ - 凯悦型复发的Analogues，用于B型和D型Eulerian多项式。我们结果的进一步推论是这些多项式的一些对称关系和$ q $ - 为B和D类型的蛇生成功能。
Carlitz and Scoville in 1973 considered a four variable polynomial that enumerates permutations in $\mathfrak{S}_n$ with respect to the parity of its descents and ascents. In recent work, Pan and Zeng proved a $q$-analogue of Carlitz-Scoville's generating function by enumerating permutations with the above four statistice along with the inversion number. Further, they also proved a type B analogue by enumerating signed permutations with respect to the parity of descents and ascents. In this work we prove a $q$-analogue of the type B result of Pan and Zeng by enumerating permutations in $\mathfrak{B}_n$ with the above four statistics and the type B inversion number. We also obtain a $q$-analogue of the generating function for the type B bivariate alternating descent polynomials. We consider a similar five-variable polynomial in the type D Coxeter groups as well and give their egf. Alternating descents for the type D groups were previously also defined by Remmel, but our definition is slightly different. As a by-product of our proofs, we get bivariate $q$-analogues of Hyatt's recurrences for the type B and type D Eulerian polynomials. Further corollaries of our results are some symmetry relations for these polynomials and $q$-analogues of generating functions for snakes of types B and D.
