Mixed Statistics on 01-Fillings of Moon Polyominoes

We establish a stronger symmetry between the numbers of northeast and southeast chains in the context of 01-fillings of moon polyominoes. Let \M be a moon polyomino with $n$ rows and $m$ columns. Consider all the 01-fillings of \M in which every row has at most one 1. We introduce four mixed statistics with respect to a bipartition of rows or columns of \M. More precisely, let $S \subseteq \{1,2,..., n\}$ and $\mathcal{R}(S)$ be the union of rows whose indices are in $S$. For any filling $M$, the top-mixed (resp. bottom-mixed) statistic $\alpha(S; M)$ (resp. $\beta(S; M)$) is the sum of the number of northeast chains whose top (resp. bottom) cell is in $\mathcal{R}(S)$, together with the number of southeast chains whose top (resp. bottom) cell is in the complement of $\mathcal{R}(S)$. Similarly, we define the left-mixed and right-mixed statistics $\gamma(T; M)$ and $\delta(T; M)$, where $T$ is a subset of the column index set $\{1,2,..., m\}$. Let $\lambda(A; M)$ be any of these four statistics $\alpha(S; M)$, $\beta(S; M)$, $\gamma(T; M)$ and $\delta(T; M)$, we show that the joint distribution of the pair $(\lambda(A; M), \lambda(\bar A; M))$ is symmetric and independent of the subsets $S, T$. In particular, the pair of statistics $(\lambda(A;M), \lambda(\bar A; M))$ is equidistributed with (\se(M),\ne(M)), where \se(M) and $\ne(M)$ are the numbers of southeast chains and northeast chains of $M$, respectively.Comment: 20 pages, 6 figure

Two-Dimensional Inversion Asymmetric Topological Insulators in Functionalized III-Bi Bilayers

The search for inversion asymmetric topological insulators (IATIs) persists as an effect for realizing new topological phenomena. However, so for only a few IATIs have been discovered and there is no IATI exhibiting a large band gap exceeding 0.6 eV. Using first-principles calculations, we predict a series of new IATIs in saturated Group III-Bi bilayers. We show that all these IATIs preserve extraordinary large bulk band gaps which are well above room-temperature, allowing for viable applications in room-temperature spintronic devices. More importantly, most of these systems display large bulk band gaps that far exceed 0.6 eV and, part of them even are up to ~1 eV, which are larger than any IATIs ever reported. The nontrivial topological situation in these systems is confirmed by the identified band inversion of the band structures and an explicit demonstration of the topological edge states. Interestingly, the nontrivial band order characteristics are intrinsic to most of these materials and are not subject to spin-orbit coupling. Owning to their asymmetric structures, remarkable Rashba spin splitting is produced in both the valence and conduction bands of these systems. These predictions strongly revive these new systems as excellent candidates for IATI-based novel applications.Comment: 17 pages,5figure