# 江苏大学夏先伟教授 ；四川大学范久瑜副教授 学术报告

 报告题目：Tree-indexed random walks报告人：夏先伟 报告时间：2020年10月21日9:00-10:30报告地点：腾讯会议ID： 875 455 377摘要：In 2007, Andrews introduced the odd   rank of  odd Durfee symbols. Let $N^{0}(m,n)$ denote the number of odd Durfee   symbols of $n$ with odd rank $m$,  and $N^{0}(r,m;n)$ be the number of   odd Durfee symbols of $n$ with odd rank congruent to $r$ modulo $m$.   Recently, Wang established explicit formulas for the generating functions of   $N^{0}(r,M;n)$ and their $8$-dissections and proved obtain some interesting   properties for $N^{0}(r,M;n)$ with $M\in \{2,4,8\}$. In this paper,we give   the generating functions for $N^{0} (r,12;n)$ by utilizing some identities   involving Appell-Lerch sums $m(x,q,z)$ and a universal mock theta function   $g(x,q)$. Based on these formulas, we determine the signs of $N^{0} (r,12;4n+t)-N^{0}(s,12;4n+t)$ for all $0\leq r, s\leq 6$ and $0\leq t \leq 3$. In particular, we prove that $(2,12;4n+1)=N^{0} (4,12;4n+1)$. Moreover,   let $\mathscr{D}_{k}^0(n)$ denote  the number of $k$-marked odd Durfee   symbols of $n$. Andrews conjectured that $\mathscr{D}_{2}^0(8n+s)$ and    $\mathscr{D}_{3}^0(16n+t)$ with $s\in \{4,6\}$ and $t\in \{1,9,11,13\}$ which were confirmed by Wang. In this paper, we found new   congruences for $\mathscr{D}_{k}^0(n)$. In particular, for $k=2$ or $3$, we   give characterizations of $n$ such that $\mathscr{D}_{k}^0(n)$ is odd and   prove that $\mathscr{D}_{k}^0(n)$ take even values with probability 1 for   $n\geq 0$. 报告人简介：夏先伟，江苏大学教授，博士生导师，江苏省杰青获得者。2010年博士毕业于南开大学，师从陈永川教授，主要研究组合数学、特殊函数与整数分拆，在包括Math. Comput., Proc. Edinb. Math. Soc.,  Pacific J. Math.,   European J. Combin., Acta Arith., J. Number Theory等国外学术期刊上发表研究论文50余篇。主持两项国家自然科学基金面上项目。 报告题目：代数组合中的牛顿多面体报告人：范久瑜报告时间：2020年10月21日10:00-1100报告地点：腾讯会议ID：875 455 377摘要：The Newton polytope of a multivariate   polynomial f is the convex hull of the exponent vectors of f. In this talk,   we will discuss the Newton polytopes of several important families of   polynomials in algebraic combinatorics, such as Schubert polynomials,   Grothendieck polynomials, key polynomials and Demazure atoms. We establish a   combinatorial algorithm to generate the vertices of a more general family of   polytopes, called Schubitopes, which include the Newton polytopes of Schubert   and key polynomials as special cases. As an application, we confirm a   conjecture of Monical, Tokcan and Yong, which asserts that the vertices of   the Newton polytope of a key polynomial can be generated by permutations in a   Bruhat order interval. This talk is based on joint work  with Peter L.   Guo.报告人简介：范久瑜，四川大学数学学院副教授，研究方向为代数组合，主要研究课题包括Schubert计数演算的组合学、对称函数、多面体的组合学等。先后主持国家自然科学青年基金、面上基金等。 邀请人：张华军