組合數學研討會
主講者: 1. 羅元勳 博士 (國立師範大學) 2. 李渭天 教授 (國立中興大學)
講題: 1.The sorting index on colored permutations and even-signed permutations 2.The Shift Operation and Intersecting Families
時間: 2014-10-17 (Fri.)  14:00 - 17:00
地點: 數學所 617 研討室 (台大院區)
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 $n$ non-attacking rooks on a fixed Ferrers shape we show that the following two sequences of set-valued statistics are joint equidistributed: $(\ell,Rmil^0,Rmil^1,\ldots,Rmil^{r-1}$, $Lmil^0,Lmil^1,\ldots,Lmil^{r-1}$, $Lmal^0,Lmal^1,\ldots,Lmal^{r-1}$, $Lmap^0,Lmap^1,\ldots,Lmap^{r-1})$ and $(sor,Cyc^0,Cyc^{r-1},\ldots,Cyc^{1}$, $Lmic^0,Lmic^{r-1},\ldots,Lmic^{1}$, $Lmal^0,Lmal^1,\ldots,Lmal^{r-1}$, $Lmap^0,Lmap^1,\ldots,Lmap^{r-1})$. We further obtain the generating function with respect to the first $2r+1$ statistics: $\sum_{\pi\in\G_{r,n,\ff}}q^{\ell(\pi)}\prod_{t=0}^{r-1}\left(\prod_{i\in Rmil^{-t}(\pi)}x_{t,i}\prod_{i\in Lmil^{-t}(\pi)}y_{t,i}\right) = \sum_{\pi\in\G_{r,n,\ff}}q^{sor(\pi)}\prod_{t=0}^{r-1}\left(\prod_{i\in Cyc^t(\pi)}x_{t,i}\prod_{i\in Lmic^t(\pi)}y_{t,i}\right) \\ =& ~~\left.\prod_{j=1}^n \right( x_{0,j}+q+\cdots+q^{j-h_j-1}+ \xi_{h_j=1}(y_{0,j})q^{j-h_j} \\ &\qquad + \left.\sum_{t=1}^{r-1} \Big(x_{r-t,j}q^{2j+t-2}+q^{2j+t-3}+\cdots+q^{j+h_j+t-1}+ \xi_{h_j=1}(y_{r-t,j})q^{j+h_j+t-2} \Big) \right).$ Analogous results are also obtained for Coxeter group of type $D$ (i.e., the even-signed permutation group). Our work generalizes recent results of Petersen[3], Chen-Gong-Guo[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.
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