题目：Bijective recurrences for Schr\"oder triangles and Comtet statistics
报告人： 傅士硕博士，重庆大学
时间：2020/6/12（周五）15：30-16：30
腾讯会议ID：648 265 758
Abstract：
Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schr\"oder paths (resp. little Schr\"oder paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. In this talk, we bijectively establish the following recurrence relations:
\begin{align*}
r(n,0)&=\sum\limits_{j=0}^{n-1}2^{j}r(n-1,j),\quad n\geq 1,\\
r(n,k)&=r(n-1,k-1)+\sum\limits_{j=k}^{n-1}2^{j-k}r(n-1,j),\quad 1\le k\le n,\\
s(n,0) &=\sum\limits_{j=1}^{n-1}2\cdot3^{j-1}s(n-1,j),\quad n\geq 1,\\
s(n,k) &=s(n-1,k-1)+\sum\limits_{j=k+1}^{n-1}2\cdot3^{j-k-1}s(n-1,j),\quad 1\le k\le n.
\end{align*}
On the other hand, it is well-known that the large Schr\"oder numbers also enumerate separable permutations. This propelled us to reveal the connection with a lesser-known permutation statistic, called initial ascending run (iar), whose distribution on separable permutations is shown to be given by $[r(n,k)]_{n,k\ge 0}$ as well. A by-product of this result is that $\iar$ is equidistributed over separable permutations with $\comp$, the number of components of a permutation. We call such statistics Comtet and we shall discuss further work concerning Comtet statistics on various classes of pattern avoiding permutations if time permits.
报告人介绍： 傅士硕，2011年博士毕业于美国宾夕法尼亚州州立大学，2011年-2012年在韩国韩国科学技术院（KAIST）从事博士后研究工作，现为重庆大学数学与统计学院特聘研究员、博士生导师。研究的主要兴趣是计数组合学、q-级数以及整数分拆理论。
