Abstract:
 1. We define a new statistic $sor$ on the set of colored permutations $\G_{r,n}$ and prove that it has the same distribution as the length function. For the set of restricted colored permutations corresponding to the arrangements of $n$ nonattacking rooks on a fixed Ferrers shape we show that the following two sequences of setvalued statistics are joint equidistributed: $(\ell,Rmil^0,Rmil^1,\ldots,Rmil^{r1}$, $Lmil^0,Lmil^1,\ldots,Lmil^{r1}$, $Lmal^0,Lmal^1,\ldots,Lmal^{r1}$, $Lmap^0,Lmap^1,\ldots,Lmap^{r1})$ and $(sor,Cyc^0,Cyc^{r1},\ldots,Cyc^{1}$, $Lmic^0,Lmic^{r1},\ldots,Lmic^{1}$, $Lmal^0,Lmal^1,\ldots,Lmal^{r1}$, $Lmap^0,Lmap^1,\ldots,Lmap^{r1})$.
We further obtain the generating function with respect to the first $2r+1$ statistics:
$\sum_{\pi\in\G_{r,n,\ff}}q^{\ell(\pi)}\prod_{t=0}^{r1}\left(\prod_{i\in Rmil^{t}(\pi)}x_{t,i}\prod_{i\in Lmil^{t}(\pi)}y_{t,i}\right) = \sum_{\pi\in\G_{r,n,\ff}}q^{sor(\pi)}\prod_{t=0}^{r1}\left(\prod_{i\in Cyc^t(\pi)}x_{t,i}\prod_{i\in Lmic^t(\pi)}y_{t,i}\right) \\
=& ~~\left.\prod_{j=1}^n \right( x_{0,j}+q+\cdots+q^{jh_j1}+ \xi_{h_j=1}(y_{0,j})q^{jh_j} \\
&\qquad + \left.\sum_{t=1}^{r1} \Big(x_{rt,j}q^{2j+t2}+q^{2j+t3}+\cdots+q^{j+h_j+t1}+ \xi_{h_j=1}(y_{rt,j})q^{j+h_j+t2} \Big) \right).$
Analogous results are also obtained for Coxeter group of type $D$ (i.e., the evensigned permutation group).
Our work generalizes recent results of Petersen[3], ChenGongGuo[1] and Poznanovic[4].
2. In this talk, we will study the shift operation which is a very important tool for solving the problems in extremal set theory. We will also see a new result obtained by Jun Wang and Huajun Zhang on the intersecting families. The result answers a question of Li, Chen, Huang and Lih posted in Electron Journal of Combinatorics, 20 (2013), # P38.
