Lecture Content：

The generating polynomial of all $n$--permutations  with respect to the number of alternating runs possesses a root at $-1$ of  multiplicity $\lfloor (n-2)/2\rfloor$ for$n\ge2$. This fact can be deduced by combining the  David--Barton formula for Eulerian polynomials with the Foata--Schützenberger $\gamma$--decomposition of these  polynomials. Recently, Bóna provided a group--action proof of this  result. In this talk, I propose an alternative approach based on the  Hetyei--Reiner action on binary trees, which yields a new combinatorial  interpretation of Bóna’s quotient polynomial. Furthermore, we  extend our study to analogous results for permutations of types~B and~D. As a  consequence of our bijective framework, we also obtain combinatorial proofs  of David--Barton type identities for permutations of types~A and~B.This talk is based on a joint work with Yunze Wang  and Jiang Zeng.

Speaker Introduction:

Qiongqiong Pan earned her Ph.D. from the Universityof Lyon in France in 2020. She joined Wenzhou University in 2021, specializingin enumerative combinatorics and orthogonal polynomial theory. Multipleresearch papers authored by her have been published in leading internationaljournals in combinatorial mathematics, such as JCTA, AAM, DM, and EJC.Currently, she is leading a Youth Project supported by the National NaturalScience Foundation of China.