The 2-color Rado number of x1+x2+⋯+xn=y1+y2+⋯+ykx_1+x_2+\cdots +x_n=y_1+y_2+\cdots +y_k

In 1982, Beutelspacher and Brestovansky determined the 2-color Rado number of the equation $$x_1+x_2+\cdots +x_{m-1}=x_m$$ for all $m\geq 3.$ Here we extend their result by determining the 2-color Rado number of the equation $$x_1+x_2+\cdots +x_n=y_1+y_2+\cdots +y_k$$ for all $n\geq 2$ and $k\geq 2.$ As a consequence, we determine the 2-color Rado number of $$x_1+x_2+\cdots +x_n=a_1y_1+\cdots +a_{\ell}y_{\ell}$$ in all cases where $n\geq 2$ and $n\geq a_1+\cdots +a_{\ell},$ and in most cases where $n\geq 2$ and $2n\geq a_1+\cdots +a_{\ell}.$Comment: 7 page

The 2-color Rado number of x1+x2+...+xm−1=axmx_1+x_2+...+x_{m-1}=ax_m

In 1982, Beutelspacher and Brestovansky proved that for every integer $m\geq 3,$ the 2-color Rado number of the equation $$x_1+x_2+...+x_{m-1}=x_m$$ is $m^2-m-1.$ In 2008, Schaal and Vestal proved that, for every $m\geq 6,$ the 2-color Rado number of $$x_1+x_2+...+x_{m-1}=2x_m$$ is $\lceil \frac{m-1}{2}\lceil\frac{m-1}{2}\rceil\rceil.$ Here we prove that, for every integer $a\geq 3$ and every $m\geq 2a^2-a+2$, the 2-color Rado number of $$x_1+x_2+...+x_{m-1}=ax_m$$ is $\lceil\frac{m-1}{a}\lceil\frac{m-1}{a}\rceil\rceil.$ For the case $a=3,$ we show that our formula gives the Rado number for all $m\geq 7,$ and we determine the Rado number for all $m\geq 3.$Comment: 15 page

The 2-color Rado Number of x1+x2+⋯+xm−1=axm,x_1+x_2+\cdots +x_{m-1}=ax_m, II

In the first installment of this series, we proved that, for every integer $a\geq 3$ and every $m\geq 2a^2-a+2$, the 2-color Rado number of $x_1 + x_2 + \cdots + x_{m-1} = ax_m$ is $\lceil\frac{m-1}{a} \lceil\frac{m-1}{a} \rceil\rceil$. Here we obtain the best possible improvement of the bound on $m.$ We prove that if $3|a$ then the 2-color Rado number is $\lceil\frac{m-1}{a} \lceil\frac{m-1}{a} \rceil\rceil$ when $m\geq 2a+1$ but not when $m=2a,$ and that if $3\nmid a$ then the 2-color Rado number is $\lceil\frac{m-1}{a} \lceil\frac{m-1}{a} \rceil\rceil$ when $m\geq 2a+2$ but not when $m=2a+1.$ We also determine the 2-color Rado number for all $a\geq 3$ and $m\geq \frac{a}{2}+1.$Comment: 18 page

On Two Bijections from S_n(321) to S_n(132)

Let S_n(321) (respectively, S_n(132)) denote the set of all permutations of {1,2,...,n} that avoid the pattern 321 (respectively, the pattern 132). Elizalde and Pak gave a bijection Theta from S_n(321) to S_n(132) that preserves the numbers of fixed points and excedances for each element of S_n(321), and commutes with the operation of taking inverses. Bloom and Saracino proved that another bijection Gamma from S_n(321) to S_n(132), introduced by Robertson, has the same properties, and they later gave a pictorial reformulation of Gamma that made these results more transparent. Here we give a pictorial reformulation of Theta, from which it follows that, although the original definitions of Theta and Gamma are very different, these two bijections are in fact related to each other in a very simple way, by using inversion, reversal, and complementation.Comment: 11 pages, 4 figure

Pattern avoidance for set partitions \`a la Klazar

In 2000 Klazar introduced a new notion of pattern avoidance in the context of set partitions of $[n]=\{1,\ldots, n\}$. The purpose of the present paper is to undertake a study of the concept of Wilf-equivalence based on Klazar's notion. We determine all Wilf-equivalences for partitions with exactly two blocks, one of which is a singleton block, and we conjecture that, for $n\geq 4$, these are all the Wilf-equivalences except for those arising from complementation. If $\tau$ is a partition of $[k]$ and $\Pi_n(\tau)$ denotes the set of all partitions of $[n]$ that avoid $\tau$, we establish inequalities between $|\Pi_n(\tau_1)|$ and $|\Pi_n(\tau_2)|$ for several choices of $\tau_1$ and $\tau_2$, and we prove that if $\tau_2$ is the partition of $[k]$ with only one block, then $|\Pi_n(\tau_1)| k$ and all partitions $\tau_1$ of $[k]$ with exactly two blocks. We conjecture that this result holds for all partitions $\tau_1$ of $[k]$. Finally, we enumerate $\Pi_n(\tau)$ for all partitions $\tau$ of $[4]$.Comment: 21 page

On criteria for rook equivalence of Ferrers boards

In [2] we introduced a new notion of Wilf equivalence of integer partitions and proved that rook equivalence implies Wilf equivalence. In the present paper we prove the converse and thereby establish a new criterion for rook equivalence. We also refine two of the standard criteria for rook equivalence and establish another new one involving what we call \emph{nested sequences of L's}.Comment: 11 pages, European Journal of Combinatorics 201

A Simple Bijective Proof of the Shape-Wilf-Equivalence of the Patterns 231 and 312

Stankova and West proved in 2002 that the patterns 231 and 312 are shape-Wilf-equivalent. Their proof was nonbijective and fairly complicated. We give a new characterization of 231 and 312 avoiding full rook placements and use this to give a simple bijective proof of the shape-Wilf- equivalence.Comment: 10 page

Practical Location Validation in Participatory Sensing Through Mobile WiFi Hotspots

The reliability of information in participatory sensing (PS) systems largely depends on the accuracy of the location of the participating users. However, existing PS applications are not able to efficiently validate the position of users in large-scale outdoor environments. In this paper, we present an efficient and scalable Location Validation System (LVS) to secure PS systems from location-spoofing attacks. In particular, the user location is verified with the help of mobile WiFi hot spots (MHSs), which are users activating the WiFi hotspot capability of their smartphones and accepting connections from nearby users, thereby validating their position inside the sensing area. The system also comprises a novel verification technique called Chains of Sight, which tackles collusion-based attacks effectively. LVS also includes a reputation-based algorithm that rules out sensing reports of location-spoofing users. The feasibility and efficiency of the WiFi-based approach of LVS is demonstrated by a set of indoor and outdoor experiments conducted using off-the-shelf smartphones, while the energy-efficiency of LVS is demonstrated by experiments using the Power Monitor energy tool. Finally, the security properties of LVS are analyzed by simulation experiments. Results indicate that the proposed LVS system is energy-efficient, applicable to most of the practical PS scenarios, and efficiently secures existing PS systems from location-spoofing attacks.Comment: IEEE TrustCom 2018, New York City, NY, US

Refined Restricted Involutions

Define $I_n^k(\alpha)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $\alpha \in S_i$, for some $i \geq 2$, and define $I_n^k(\emptyset;\alpha)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which contain the pattern $\alpha \in S_i$, for some $i \geq 2$, exactly once. Let $i_n^k(\alpha)$ be the number of elements in $I_n^k(\alpha)$ and let $i_n^k(\emptyset;\alpha)$ be the number of elements in $I_n^k(\emptyset;\alpha)$. We investigate $I_n^k(\alpha)$ and $I_n^k(\emptyset;\alpha)$ for all $\alpha \in S_3$. In particular, we show that $i_n^k(132)=i_n^k(213)=i_n^k(321)$, $i_n^k(231)=i_n^k(312)$, $i_n^k(\emptyset;132) =i_n^k(\emptyset;213)$, and $i_n^k(\emptyset;231)=i_n^k(\emptyset;312)$ for all $0 \leq k \leq n$.Comment: 20 page

Cell List Algorithms for Nonequilibrium Molecular Dynamics

We present two modifications of the standard cell list algorithm for nonequilibrium molecular dynamics simulations of homogeneous, linear flows. When such a flow is modeled with periodic boundary conditions, the simulation box deforms with the flow, and recent progress has been made developing boundary conditions suitable for general 3D flows of this type. For the typical case of short-ranged, pairwise interactions, the cell list algorithm reduces computational complexity of the force computation from O($N^2$) to O($N$), where $N$ is the total number of particles in the simulation box. The new versions of the cell list algorithm handle the dynamic, deforming simulation geometry. We include a comparison of the complexity and efficiency of the two proposed modifications of the standard algorithm.Comment: 13 pages, 10 figure