1. We define a new statistic on the set of colored permutations and prove that it has the same distribution as the length function. For the set of restricted colored permutations corresponding to the arrangements of non-attacking rooks on a fixed Ferrers shape we show that the following two sequences of set-valued statistics are joint equidistributed: , , , and , , , .
We further obtain the generating function with respect to the first statistics:
Analogous results are also obtained for Coxeter group of type (i.e., the even-signed permutation group).
Our work generalizes recent results of Petersen, Chen-Gong-Guo and Poznanovic.
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.|