The maximum drop size of a permutation $pi$ of $[n]={1,2,ldots, n}$ is defined to be the maximum value of $i-pi(i)$. Chung, Claesson, Dukes and Graham obtained polynomials $P_k(x)$ that can be used to determine the number of permutations of $[n]$ wit
h $d$ descents and maximum drop size not larger than $k$. Furthermore, Chung and Graham gave combinatorial interpretations of the coefficients of $Q_k(x)=x^k P_k(x)$ and $R_{n,k}(x)=Q_k(x)(1+x+cdots+x^k)^{n-k}$, and raised the question of finding a bijective proof of the symmetry property of $R_{n,k}(x)$. In this paper, we establish a bijection $varphi$ on $A_{n,k}$, where $A_{n,k}$ is the set of permutations of $[n]$ and maximum drop size not larger than $k$. The map $varphi$ remains to be a bijection between certain subsets of $A_{n,k}$. %related to the symmetry property. This provides an answer to the question of Chung and Graham. The second result of this paper is a proof of a conjecture of Hyatt concerning the unimodality of polynomials in connection with the number of signed permutations of $[n]$ with $d$ type $B$ descents and the type $B$ maximum drop size not greater than $k$.
An alternating permutation of length $n$ is a permutation $pi=pi_1 pi_2 ... pi_n$ such that $pi_1 < pi_2 > pi_3 < pi_4 > ...$. Let $A_n$ denote set of alternating permutations of ${1,2,..., n}$, and let $A_n(sigma)$ be set of alternating permutations
in $A_n$ that avoid a pattern $sigma$. Recently, Lewis used generating trees to enumerate $A_{2n}(1234)$, $A_{2n}(2143)$ and $A_{2n+1}(2143)$, and he posed several conjectures on the Wilf-equivalence of alternating permutations avoiding certain patterns. Some of these conjectures have been proved by Bona, Xu and Yan. In this paper, we prove the two relations $|A_{2n+1}(1243)|=|A_{2n+1}(2143)|$ and $|A_{2n}(4312)|=|A_{2n}(1234)|$ as conjectured by Lewis.
We propose a second renormalization group method to handle the tensor-network states or models. This method reduces dramatically the truncation error of the tensor renormalization group. It allows physical quantities of classical tensor-network model
s or tensor-network ground states of quantum systems to be accurately and efficiently determined.
We investigate the crystallization of amorphous arsenic-selenium alloys with 0%, 0.5%, 2%, 6%, 10%, and 19% arsenic by atomic concentration using synchrotron X-ray absorption spectroscopy. We identify crystalline order using the extended X-ray absorp
tion fine structure (EXAFS) spectra and correlate this order to changes in features of the X-ray absorption near edge structure (XANES) spectra. We find supporting evidence that the structure of amorphous selenium is composed of disordered helical chains, and is therefore closer to the trigonal crystalline phase than the monoclinic crystalline phase.
The radiative decay of quantum dot (QD) excitons into surface plasmons in a cylindrical nanowire is investigated theoretically. Maxwells equations with appropriate boundary conditions are solved numerically to obtain the dispersion relations of surfa
ce plasmons. The radiative decay rate of QD excitons is found to be greatly enhanced at certain values of the exciton bandgap. Analogous to the decay of a two-level atom in the photonic crystal, we first point out that such an enhanced phenomenon allows one to examine the non-Markovian dynamics of the QD exciton. Besides, due to the one dimensional propagating feature of nanowire surface plasmons, remote entangled states can be generated via super-radiance and may be useful in future quantum information processing.
