The Weak Bruhat Order and Separable Permutations

In this paper we consider the rank generating function of a separable permutation $\pi$ in the weak Bruhat order on the two intervals $[\text{id}, \pi]$ and $[\pi, w_0]$, where $w_0 = n,(n-1),..., 1$. We show a surprising result that the product of these two generating functions is the generating function for the symmetric group with the weak order. We then obtain explicit formulas for the rank generating functions on $[\text{id}, \pi]$ and $[\pi, w_0]$, which leads to the rank-symmetry and unimodality of the two graded posets

Effects of center offset and noise on weak-lensing derived concentration-mass relation of dark matter halos

With the halo catalog from the {\it Millennium Simulation}, we analyze the weak-lensing measured density profiles for clusters of galaxies, paying attention to the determination of the concentration-mass ($c$-$M$) relation which can be biased by the center offset, selection effect, and shape noise from intrinsic ellipticities of background galaxies. Several different methods of locating the center of a cluster from weak-lensing effects alone are explored. We find that, for intermediate redshift clusters, the highest peak from our newly proposed two-scale smoothing method applied to the reconstructed convergence field, first with a smoothing scale of $2^\prime$ and then $0\overset{\prime}{.}5$, corresponds best to the true center. Assuming the parameterized Navarro-Frenk-White profile, we fit the reduced tangential shear signals around different centers identified by different methods. It is shown that, for the ensemble median values, a center offset larger than one scale radius $r_s$ can bias the derived mass and concentration significantly lower than the true values, especially for low-mass halos. However, the existence of noise can compensate for the offset effect and reduce the systematic bias, although the scatter of mass and concentration becomes considerably larger. Statistically, the bias effect of center offset on the $c$-$M$ relation is insignificant if an appropriate center finding method is adopted. On the other hand, noise from intrinsic ellipticities can bias the $c$-$M$ relation derived from a sample of weak-lensing analyzed clusters if a simple $\chi^2$ fitting method is used. To properly account for the scatter and covariance between $c$ and $M$, we apply a Bayesian method to improve the statistical analysis of the $c$-$M$ relation. It is shown that this new method allows us to derive the $c$-$M$ relation with significantly reduced biases.Comment: Accepted for Publication in ApJ. 25 pages, 14 figures. Updated to match the published versio