http://www.combinatorics.org/ojs/index.php/eljc/issue/feedThe Electronic Journal of Combinatorics2014-09-18T10:45:48+10:00André Kündgenakundgen@csusm.eduOpen Journal Systems<p>The copyright of published papers remains with the authors. We only require your agreement that we publish it, as described in the following publication release agreement:</p><ol><li>This is an agreement between the Electronic Journal of Combinatorics (the "Journal"), and the copyright owner (the "Owner") of a work (the "Work") to be published in the Journal.</li><li>The Owner warrants that s/he has the full power and authority to enter into this Agreement and to grant the rights granted in this Agreement.</li><li>The Owner hereby grants to the Journal a worldwide, irrevocable, royalty free license to publish or distribute the Work, to enter into arrangements with others to publish or distribute the Work, and to archive the Work.</li><li>The Owner agrees that further publication of the Work, with the same or substantially the same content as appears in the Journal, will include an acknowledgement of prior publication in the Journal.</li></ol><p>The Electronic Journal of Combinatorics (E-JC) is a fully-refereed electronic journal with <em>very high standards,</em> publishing papers of substantial content and interest in all branches of discrete mathematics, including combinatorics, graph theory, and algorithms for combinatorial problems. The journal is <em>completely free</em> for both authors and readers. Authors retain the copyright of their articles. Articles published in E-JC are reviewed on MathSciNet and ZBMath, and are indexed by Web of Science. The latest papers are always available by clicking on the tab marked "Current" near the top of the page. You can also locate papers using the search facility on the right hand side of this page.</p><p>E-JC was founded in 1994 by Herbert S. Wilf and Neil Calkin, making it one of the oldest electronic journals.</p>http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p1A Spectral Equivalent Condition of the $P$-Polynomial Property for Association Schemes2014-08-21T10:24:22+10:00Hiroshi Nozakihnozaki@auecc.aichi-edu.ac.jpHirotake Kuriharakurihara@kct.ac.jpWe give two equivalent conditions of the $P$-polynomial property of a symmetric association scheme. The first equivalent condition shows that the $P$-polynomial property is determined only by the first and second eigenmatrices of the symmetric association scheme. The second equivalent condition is another expression of the first using predistance polynomials.2014-07-03T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p2Consecutive Up-Down Patterns in Up-Down Permutations2014-08-21T10:24:22+10:00Jeffrey B. Remmeljremmel@ucsd.edu<p>In this paper, we study the distribution of the number of consecutive pattern matches of the five up-down permutations of length four, $1324$, $2314$, $2413$, $1432$, and $3412$, in the set of up-down permutations. We show that for any such $\tau$, the generating function for the distribution of the number of consecutive pattern matches of $\tau$ in the set of up-down permutations can be expressed in terms of what we call the generalized maximum packing polynomials of $\tau$. We then provide some systematic methods to compute the generalized maximum packing polynomials for such $\tau$.</p>2014-07-03T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p3A Characteristic Factor for the 3-Term IP Roth Theorem in $\mathbb{Z}_3^\mathbb{N}$2014-08-21T10:24:22+10:00Randall McCutcheonrmcctchn@memphis.eduAlistair Windsorawindsor@memphis.edu<p>Let $\Omega = \bigoplus_{i=1}^\infty \mathbb{Z}_3$ and $e_i = (0, \dots, 0 , 1, 0, \dots)$ where the $1$ occurs in the $i$-th coordinate. Let $\mathscr{F}=\{ \alpha \subset \mathbb{N} : \varnothing \neq \alpha, \alpha \text{ is finite} \}$. <span style="font-size: 10px;">There is a natural inclusion of $\mathscr{F}$ into $\Omega$ where $\alpha \in \mathscr{F}$ is mapped to $e_\alpha = \sum_{i \in \alpha} e_i$. We give a new proof that if $E \subset \Omega$ with $d^*(E) >0$ then there exist $\omega \in \Omega$ and $\alpha \in \mathscr{F}$ such that \[ </span><span style="font-size: 10px;">\{ \omega, \omega+ e_\alpha, \omega + 2 e_\alpha \} \subset E.\]</span><span style="font-size: 10px;">Our proof establishes that for the ergodic reformulation of the problem there is a characteristic factor that is a one step compact extension of the Kronecker factor.</span></p>2014-07-03T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p4On Fence Patrolling by Mobile Agents2014-08-21T10:24:22+10:00Adrian Dumitrescudumitres@uwm.eduAnirban Ghoshanirban@uwm.eduCsaba D. Tóthcdtoth@acm.orgSuppose that a fence needs to be protected (perpetually) by $k$ mobile agents with maximum speeds $v_1,\ldots,v_k$ so that no point on the fence is left unattended for more than a given amount of time. The problem is to determine if this requirement can be met, and if so, to design a suitable patrolling schedule for the agents. Alternatively, one would like to find a schedule that minimizes the <em>idle time</em>, that is, the longest time interval during which some point is not visited by any agent. We revisit this problem, introduced by Czyzowicz et al. (2011), and discuss several strategies for the cases where the fence is an open and a closed curve, respectively.<br /><br />In particular: (i) we disprove a conjecture by Czyzowicz et al. regarding the optimality of their algorithm ${\mathcal A}_2$ for unidirectional patrolling of a closed fence; (ii) we present a schedule with a lower idle time for patrolling an open fence, improving an earlier result of Kawamura and Kobayashi.<br /><br /><br /><br />2014-07-10T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p5Arc-Transitive Dihedral Regular Covers of Cubic Graphs2014-08-21T10:24:22+10:00Jicheng Mama_jicheng@hotmail.com<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>A regular covering projection is called </span><span><em>dihedral</em> </span><span>or </span><span><em>abelian</em> </span><span>if the covering transformation group is dihedral or abelian. A lot of work has been done with regard to the classification of arc-transitive abelian (or elementary abelian, or cyclic) covers of symmetric graphs. In this paper, we investigate arc-transitive dihedral regular covers of symmetric (arc-transitive) cubic graphs. In particular, we classify all arc-transitive dihedral regular covers of $</span><span>K_</span><span>4$</span><span>, $</span><span>K_{</span><span>3</span><span>,</span><span>3}$</span><span>, the 3-cube $</span><span>Q_</span><span>3$ </span><span>and the Petersen graph.</span></p></div></div></div>2014-07-10T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p6Nordhaus-Gaddum Type Inequalities for Laplacian and Signless Laplacian Eigenvalues2014-08-21T10:24:22+10:00F. Ashraffirouzeh_ashraf@yahoo.comB. Tayfeh-Rezaietayfeh-r@ipm.ir<p>Let $G$ be a graph with $n$ vertices. We denote the largest signless Laplacian eigenvalue of $G$ by $q_1(G)$ and Laplacian eigenvalues of $G$ by $\mu_1(G)\ge\cdots\ge\mu_{n-1}(G)\ge\mu_n(G)=0$. It is a conjecture on Laplacian spread of graphs that $\mu_1(G)-\mu_{n-1}(G)\le n-1$ or equivalently $\mu_1(G)+\mu_1(\overline G)\le2n-1$. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph $G$, $\mu_1(G)\mu_1(\overline G)\le n(n-1)$. Aouchiche and Hansen [<em>Discrete Appl. Math.</em> 2013] conjectured that $q_1(G)+q_1(\overline G)\le3n-4$ and $q_1(G)q_1(\overline G)\le2n(n-2)$. We prove the former and disprove the latter by constructing a family of graphs $H_n$ where $q_1(H_n)q_1(\overline{H_n})$ is about $2.15n^2+O(n)$.</p>2014-07-10T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p7The Minimum Number of Nonnegative Edges in Hypergraphs2014-08-21T10:24:22+10:00Hao Huanghuanghao@math.ias.eduBenny Sudakovbenjamin.sudakov@math.ethz.ch<p>An $r$-uniform $n$-vertex hypergraph $H$ is said to have the Manickam-Miklós-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of $H$. In this paper we show that for $n>10r^3$, every $r$-uniform $n$-vertex hypergraph with equal codegrees has the MMS property, and the bound on $n$ is essentially tight up to a constant factor. This result has two immediate corollaries. First it shows that every set of $n>10k^3$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ nonnegative $k$-sums, verifying the Manickam-Miklós-Singhi conjecture for this range. More importantly, it implies the vector space Manickam-Miklós-Singhi conjecture which states that for $n \ge 4k$ and any weighting on the $1$-dimensional subspaces of $\mathbb{F}_{q}^n$ with nonnegative sum, the number of nonnegative $k$-dimensional subspaces is at least ${n-1 \brack k-1}_q$. We also discuss two additional generalizations, which can be regarded as analogues of the <span>Erdős</span><span>-</span><span>Ko</span><span>-</span><span>Rado </span>theorem on $k$-intersecting families.</p>2014-07-10T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p8On the Cayley Isomorphism Problem for Cayley Objects of Nilpotent Groups of Some Orders2014-08-21T10:24:22+10:00Edward Dobsondobson@math.msstate.edu<p>We give a necessary condition to reduce the Cayley isomorphism problem for Cayley objects of a nilpotent or abelian group $G$ whose order satisfies certain arithmetic properties to the Cayley isomorphism problem of Cayley objects of the Sylow subgroups of $G$ in the case of nilpotent groups, and in the case of abelian groups to certain natural subgroups. As an application of this result, we show that ${\mathbb Z}_q\times{\mathbb Z}_p^2\times{\mathbb Z}_m$ is a CI-group with respect to digraphs, where $q$ and $p$ are primes with $p^2 < q$ and $m$ is a square-free integer satisfying certain arithmetic conditions (but there are no other restrictions on $q$ and $p$).</p>2014-07-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p9Ascent-Descent Young Diagrams and Pattern Avoidance in Alternating Permutations2014-08-21T10:24:22+10:00Ravi Jagadeesanravi.jagadeesan@gmail.com<p><span style="color: #000000;">We investigate pattern avoidance in alternating permutations and an alternating analogue of Young diagrams. </span><span style="color: #000000;">In particular, using an extension of </span><span style="color: #000000; text-decoration: underline;">Babson</span><span style="color: #000000;"> and West's notion of shape-</span><span style="color: #000000; text-decoration: underline;">Wilf</span><span style="color: #000000;"> equivalence </span><span style="color: #000000;">described in our recent paper (with N. </span><span style="color: #000000; text-decoration: underline;">Gowravaram</span><span style="color: #000000;">), we generalize results of </span><span style="color: #000000; text-decoration: underline;">Backelin</span><span style="color: #000000;">, West, and </span><span style="color: #000000; text-decoration: underline;">Xin </span><span style="color: #000000;">and </span><span style="color: #000000; text-decoration: underline;">Ouchterlony </span><span style="color: #000000;">to alternating permutations. Unlike </span><span style="color: #000000; text-decoration: underline;">Ouchterlony</span><span style="color: #000000;"> and Bóna</span><span style="color: #000000;">'s </span><span style="color: #000000; text-decoration: underline;">bijections</span><span style="color: #000000;">, our </span><span style="color: #000000; text-decoration: underline;">bijections</span><span style="color: #000000;"> are not the restrictions of </span><span style="color: #000000; text-decoration: underline;">Backelin</span><span style="color: #000000;">, West, and </span><span style="color: #000000; text-decoration: underline;">Xin's </span><span style="color: #000000; text-decoration: underline;">bijections</span><span style="color: #000000;"> to alternating permutations. </span><span style="color: #000000;">This paper is the second of a two-paper series presenting the work of </span><span style="color: #000000;"><em>Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux </em>(with N. </span><span style="color: #000000; text-decoration: underline;">Gowravaram</span><span style="color: #000000;">, <a href="http://arxiv.org/abs/1301.6796v1">arXiv:</a></span><a href="http://arxiv.org/abs/1301.6796v1"><span style="color: #000000;">1301.</span><span style="color: #000000; text-decoration: underline;">6796v1</span></a><span style="color: #000000;">). The first</span><span style="color: #000000;"> paper in the series is </span><span style="color: #000000;"><em>Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux</em> (with N. </span><span style="color: #000000; text-decoration: underline;">Gowravaram</span><span style="color: #000000;">, </span><span style="color: #000000;"><em>Electronic Journal of </em></span><em><span style="color: #000000;">Combinatorics </span></em><a href="/ojs/index.php/eljc/article/view/v20i4p17" target="_blank"><span style="color: #000000;">20(4):#</span></a><span style="color: #800000;"><a href="/ojs/index.php/eljc/article/view/v20i4p17" target="_blank">P17</a>,</span><span style="color: #000000;"> 2013).</span><span style="color: #000000;"><br /></span></p>2014-07-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p10The Expected Characteristic and Permanental Polynomials of the Random Gram Matrix2014-08-21T10:24:22+10:00Jacob G. Martinjm733@georgetown.eduE. Rodney Canfielderc@cs.uga.edu<p>A $t \times n$ random matrix $A$ can be formed by sampling $n$ independent random column vectors, each containing $t$ components. The <em>random Gram matrix</em> of size $n$, $G_{n}=A^{T}A$, contains the dot products between all pairs of column vectors in the randomly generated matrix $A$, and has characteristic roots coinciding with the singular values of $A$. Furthermore, the sequences $\det{(G_{i})}$ and $\text{perm}(G_{i})$ (for $i = 0, 1, \dots, n$) are factors that comprise the expected coefficients of the characteristic and permanental polynomials of $G_{n}$. We prove theorems that relate the generating functions and recursions for the traces of matrix powers, expected characteristic coefficients, expected determinants $E(\det{(G_{n})})$, and expected permanents $E(\text{perm}(G_{n}))$ in terms of each other. Using the derived recursions, we exhibit the efficient computation of the expected determinant and expected permanent of a random Gram matrix $G_{n}$, formed according to any underlying distribution. These theoretical results may be used both to speed up numerical algorithms and to investigate the numerical properties of the expected characteristic and permanental coefficients of any matrix comprised of independently sampled columns.</p>2014-07-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p11A Counterexample to a Question of Hof, Knill and Simon2014-08-21T10:24:22+10:00Sébastien Labbélabbe@liafa.univ-paris-diderot.fr<p>In this article, we give a negative answer to a question of Hof, Knill and Simon (1995) concerning purely morphic sequences obtained from primitive morphism containing an infinite number of palindromes. Their conjecture states that such palindromic sequences arise from substitutions that are in class $\mathcal{P}$. The conjecture was proven for the binary alphabet by B. Tan in 2007. We give here a counterexample for a ternary alphabet.<br /><br /></p>2014-07-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p12Counting Results for Thin Butson Matrices2014-08-21T10:24:22+10:00Teo Banicateo.banica@gmail.comA partial Butson matrix is a matrix $H\in M_{M\times N}(\mathbb Z_q)$ having its rows pairwise orthogonal, where $\mathbb Z_q\subset\mathbb C^\times$ is the group of $q$-th roots of unity. We investigate here the counting problem for these matrices in the "thin" regime, where $M=2,3,\ldots$ is small, and where $N\to\infty$ (subject to the condition $N\in p\mathbb N$ when $q=p^k>2$). The proofs are inspired from the de Launey-Levin and Richmond-Shallit counting results.2014-07-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p13Trivial Meet and Join within the Lattice of Monotone Triangles2014-08-21T10:24:22+10:00John Engbersjohnengbers@gmail.comAdam Hammettadam.hammett@bethelcollege.eduThe lattice of monotone triangles $(\mathfrak{M}_n,\leq)$ ordered by entry-wise comparisons is studied. Let $\tau_{\min}$ denote the unique minimal element in this lattice, and $\tau_{\max}$ the unique maximum. The number of $r$-tuples of monotone triangles $(\tau_1,\ldots,\tau_r)$ with minimal infimum $\tau_{\min}$ (maximal supremum $\tau_{\max}$, resp.) is shown to asymptotically approach $r|\mathfrak{M}_n|^{r-1}$ as $n \to \infty$. Thus, with high probability this event implies that one of the $\tau_i$ is $\tau_{\min}$ ($\tau_{\max}$, resp.). Higher-order error terms are also discussed.2014-07-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p14On the Strong Partition Dimension of Graphs2014-08-21T10:24:22+10:00Ismael González Yeroismael.gonzalez@uca.es<p>We present a new style of metric generator in graphs. Specifically we introduce a metric generator based on a partition of the vertex set of a graph. The sets of the partition will work as the elements which will uniquely determine the position of each single vertex of the graph. A set $W$ of vertices of a connected graph $G$ strongly resolves two different vertices $x,y\notin W$ if either $d_G(x,W)=d_G(x,y)+d_G(y,W)$ or $d_G(y,W)=d_G(y,x)+d_G(x,W)$, where $d_G(x,W)=\min\left\{d(x,w)\;:\;w\in W\right\}$. An ordered vertex partition $\Pi=\left\{U_1,U_2,...,U_k\right\}$ of a graph $G$ is a strong resolving partition for $G$ if every two different vertices of $G$ belonging to the same set of the partition are strongly resolved by some set of $\Pi$. A strong resolving partition of minimum cardinality is called a strong partition basis and its cardinality the strong partition dimension. In this article we introduce the concepts of strong resolving partition and strong partition dimension and we begin with the study of its mathematical properties.</p>2014-07-25T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p15The Weak Order on Pattern-Avoiding Permutations2014-08-21T10:24:22+10:00Brian Drakedrakebr@gvsu.eduThe weak order on the symmetric group is a well-known partial order which is also a lattice. We consider subposets of the weak order consisting of permutations avoiding a single pattern, characterizing the patterns for which the subposet is a lattice. These patterns have only a single small ascent or descent. We prove that all patterns for which the subposet is a sublattice have length at most three.2014-07-25T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p16An Erdős-Ko-Rado Theorem for Permutations with Fixed Number of Cycles2014-08-21T10:24:22+10:00Cheng Yeaw Kumatkcy@nus.edu.sgKok Bin Wongkbwong@um.edu.my<p>Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For a positive integer $k$, define $S_{n,k}$ to be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles, i.e.,<br />\[ S_{n,k} = \{\pi \in S_{n}: \pi = c_{1}c_{2} \cdots c_{k}\},\] <br />where $c_1,c_2,\dots ,c_k$ are disjoint cycles. The size of $S_{n,k}$ is $\left [ \begin{matrix}n\\ k \end{matrix}\right]=(-1)^{n-k}s(n,k)$, where $s(n,k)$ is the Stirling number of the first kind. A family $\mathcal{A} \subseteq S_{n,k}$ is said to be $t$-<em>cycle-intersecting</em> if any two elements of $\mathcal{A}$ have at least $t$ common cycles. In this paper we show that, given any positive integers $k,t$ with $k\geq t+1$, if $\mathcal{A} \subseteq S_{n,k}$ is $t$-cycle-intersecting and $n\ge n_{0}(k,t)$ where $n_{0}(k,t) = O(k^{t+2})$, then <br />\[ |\mathcal{A}| \le \left [ \begin{matrix}n-t\\ k-t \end{matrix}\right],\]<br />with equality if and only if $\mathcal{A}$ is the stabiliser of $t$ fixed points.</p>2014-07-25T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p17Some Identities involving the Partial Sum of $q$-Binomial Coefficients2014-08-21T10:24:22+10:00Bing Heyuhe001@foxmail.comWe give some identities involving sums of powers of the partial sum of $q$-binomial coefficients, which are $q$-analogues of Hirschhorn's identities [<em>Discrete Math.</em> 159 (1996), 273-278] and Zhang's identity [<em>Discrete Math.</em> 196 (1999), 291-298].2014-07-25T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p18Minimum-Weight Edge Discriminators in Hypergraphs2014-08-21T10:24:22+10:00Bhaswar B. Bhattacharyabhaswar.bhattacharya@gmail.comSayantan Dassayantan@umich.eduShirshendu Gangulysganguly@math.washington.eduIn this paper we introduce the notion of minimum-weight edge-discriminators in hypergraphs, and study their various properties. For a hypergraph $\mathcal H=(\mathcal V, \mathscr E)$, a function $\lambda: \mathcal V\rightarrow \mathbb Z^{+}\cup\{0\}$ is said to be an <em>edge-discriminator</em> on $\mathcal H$ if $\sum_{v\in E_i}{\lambda(v)}>0$, for all hyperedges $E_i\in \mathscr E$, and $\sum_{v\in E_i}{\lambda(v)}\ne \sum_{v\in E_j}{\lambda(v)}$, for every two distinct hyperedges $E_i, E_j \in \mathscr E$. An <em>optimal edge-discriminator</em> on $\mathcal H$, to be denoted by $\lambda_\mathcal H$, is an edge-discriminator on $\mathcal H$ satisfying $\sum_{v\in \mathcal V}\lambda_\mathcal H (v)=\min_\lambda\sum_{v\in \mathcal V}{\lambda(v)}$, where the minimum is taken over all edge-discriminators on $\mathcal H$. We prove that any hypergraph $\mathcal H=(\mathcal V, \mathscr E)$, with $|\mathscr E|=m$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H(v)\leq m(m+1)/2$, and the equality holds if and only if the elements of $\mathscr E$ are mutually disjoint. For $r$-uniform hypergraphs $\mathcal H=(\mathcal V, \mathscr E)$, it follows from earlier results on Sidon sequences that $\sum_{v\in \mathcal V}\lambda_{\mathcal H}(v)\leq |\mathcal V|^{r+1}+o(|\mathcal V|^{r+1})$, and the bound is attained up to a constant factor by the complete $r$-uniform hypergraph. Finally, we show that no optimal edge-discriminator on any hypergraph $\mathcal H=(\mathcal V, \mathscr E)$, with $|\mathscr E|=m~(\geq 3)$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H (v)=m(m+1)/2-1$. This shows that all integer values between $m$ and $m(m+1)/2$ cannot be the weight of an optimal edge-discriminator of a hypergraph, and this raises many other interesting combinatorial questions.<br /><br /><br /><br />2014-08-06T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p19The Maximal Length of a $k$-Separator Permutation2014-08-21T10:24:22+10:00Benjamin Gunbybgunby314@gmail.comA permutation $\sigma\in S_n$ is a $k$-separator if all of its patterns of length $k$ are distinct. Let $F(k)$ denote the maximal length of a $k$-separator. Hegarty (2013) showed that $k+\left\lfloor\sqrt{2k-1}\right\rfloor-1\leq F(k)\leq k+\left\lfloor\sqrt{2k-3}\right\rfloor$, and conjectured that $F(k)=k+\left\lfloor\sqrt{2k-1}\right\rfloor-1$. This paper will strengthen the upper bound to prove the conjecture for all sufficiently large $k$ (in particular, for all $k\geq 320801$).2014-08-06T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p20Grid Minors in Damaged Grids2014-08-21T10:24:22+10:00David Eppsteindavid.eppstein@gmail.com<p>We prove upper and lower bounds on the size of the largest square grid graph that is a subgraph, minor, or shallow minor of a graph in the form of a larger square grid from which a specified number of vertices have been deleted. Our bounds are tight to within constant factors. We also provide less-tight bounds on analogous problems for higher-dimensional grids.</p>2014-08-06T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p21Extremal Problems for the $p$-Spectral Radius of Graphs2014-08-21T10:24:22+10:00Liying Kanglykang@shu.edu.cnVladimir Nikiforovvnikifrv@memphis.edu<p>The $p$-spectral radius of a graph $G\ $of order $n$ is defined for any real number $p\geq1$ as</p><p>$$\lambda^{(p)}(G) =\max\{ 2\sum_{\{i,j\}\in E(G)} x_ix_j:x_1,\ldots,x_n\in\mathbb{R}\text{ and }\vert x_{1}\vert ^{p}+\cdots+\vert x_n\vert^{p}=1\} .$$</p><p>The most remarkable feature of $\lambda^{(p)}$ is that it seamlessly joins several other graph parameters, e.g., $\lambda^{(1)}$ is the Lagrangian, $\lambda^{(2) }$ is the spectral radius and $\lambda^{(\infty) }/2$ is the number of edges. This paper presents solutions to some extremal problems about $\lambda^{(p)}$, which are common generalizations of corresponding edge and spectral extremal problems.</p><p>Let $T_{r}\left( n\right) $ be the $r$-partite Turán<strong> </strong>graph of order $n$. Two of the main results in the paper are:</p><p>(I) Let $r\geq2$ and $p>1.$ If $G$ is a $K_{r+1}$-free graph of order $n$, then<br />$$\lambda^{(p)}(G) <\lambda^{(p)}(T_{r}(n)),$$ unless $G=T_{r}(n)$.</p><p>(II) Let $r\geq2$ and $p>1.$ If $G\ $is a graph of order $n,$ with</p><p>$$\lambda^{(p)}(G)>\lambda^{(p)}( T_{r}(n)) ,$$</p><p><br />then $G$ has an edge contained in at least $cn^{r-1}$ cliques of order $r+1$, where $c$ is a positive number depending only on $p$ and $r.$</p>2014-08-06T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p22On Saturated $k$-Sperner Systems2014-08-21T10:24:22+10:00Natasha Morrisonmorrison@maths.ox.ac.ukJonathan A. Noelnoel@maths.ox.ac.ukAlex Scottscott@maths.ox.ac.uk<p>Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to be $k$<em>-Sperner</em> if it does not contain a chain of length $k+1$ under set inclusion and it is <em>saturated</em> if it is maximal with respect to this property. Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $\varepsilon>0$ such that for every $k$ and $|X| \geq n_0(k)$ there exists a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ with cardinality at most $2^{(1-\varepsilon)k}$.</p><p>A collection $\mathcal{F}\subseteq \mathcal{P}(X)$ is said to be an <em>oversaturated</em> $k$<em>-Sperner system</em> if, for every $S\in\mathcal{P}(X)\setminus\mathcal{F}$, $\mathcal{F}\cup\{S\}$ contains more chains of length $k+1$ than $\mathcal{F}$. Gerbner et al. proved that, if $|X|\geq k$, then the smallest such collection contains between $2^{k/2-1}$ and $O\left(\frac{\log{k}}{k}2^k\right)$ elements. We show that if $|X|\geq k^2+k$, then the lower bound is best possible, up to a polynomial factor.</p><pre><!--EndFragment--></pre>2014-08-13T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p23On the Number of Colored Birch and Tverberg Partitions2014-08-21T10:24:22+10:00Stephan Hellstephan@hell-wie-dunkel.de<p>In 2009, Blagojević, Matschke, and Ziegler established the first tight colored Tverberg theorem. We develop a colored version of our previous results (2008): Evenness and non-trivial lower bounds for the number of colored Tverberg partitions. Both properties follow from similar results on the number of colored Birch partitions.</p>2014-08-13T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p24Bounding Sequence Extremal Functions with Formations2014-08-21T10:24:22+10:00Jesse Genesongeneson@math.mit.eduRohil Prasadprasad01@college.harvard.eduJonathan Tidorjtidor@mit.edu<p>An $(r, s)$-formation is a concatenation of $s$ permutations of $r$ letters. If $u$ is a sequence with $r$ distinct letters, then let $\mathit{Ex}(u, n)$ be the maximum length of any $r$-sparse sequence with $n$ distinct letters which has no subsequence isomorphic to $u$. For every sequence $u$ define $\mathit{fw}(u)$, the formation width of $u$, to be the minimum $s$ for which there exists $r$ such that there is a subsequence isomorphic to $u$ in every $(r, s)$-formation. We use $\mathit{fw}(u)$ to prove upper bounds on $\mathit{Ex}(u, n)$ for sequences $u$ such that $u$ contains an alternation with the same formation width as $u$.</p><p>We generalize Nivasch's bounds on $\mathit{Ex}((ab)^{t}, n)$ by showing that $\mathit{fw}((12 \ldots l)^{t})=2t-1$ and $\mathit{Ex}((12\ldots l)^{t}, n) =n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2}\pm O(\alpha(n)^{t-3})}$ for every $l \geq 2$ and $t\geq 3$, such that $\alpha(n)$ denotes the inverse Ackermann function. Upper bounds on $\mathit{Ex}((12 \ldots l)^{t} , n)$ have been used in other papers to bound the maximum number of edges in $k$-quasiplanar graphs on $n$ vertices with no pair of edges intersecting in more than $O(1)$ points.</p><p>If $u$ is any sequence of the form $a v a v' a$ such that $a$ is a letter, $v$ is a nonempty sequence excluding $a$ with no repeated letters and $v'$ is obtained from $v$ by only moving the first letter of $v$ to another place in $v$, then we show that $\mathit{fw}(u)=4$ and $\mathit{Ex}(u, n) =\Theta(n\alpha(n))$. Furthermore we prove that $\mathit{fw}(abc(acb)^{t})=2t+1$ and $\mathit{Ex}(abc(acb)^{t}, n) = n2^{\frac{1}{(t-1)!}\alpha(n)^{t-1}\pm O(\alpha(n)^{t-2})}$ for every $t\geq 2$.</p>2014-08-13T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p25Counting the Palstars2014-08-21T10:24:22+10:00L. Bruce Richmondlbrichmo@uwaterloo.caJeffrey O Shallitshallit@cs.uwaterloo.ca<p>A palstar (after Knuth, Morris, and Pratt) is a concatenation of even-length palindromes. We show that, asymptotically, there are $\Theta(\alpha_k^n)$ palstars of length $2n$ over a $k$-letter alphabet, where $\alpha_k$ is a constant such that $2k-1 < \alpha_k < 2k-{1 \over 2}$. In particular, $\alpha_2\doteq 3.33513193$.</p>2014-08-13T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p26A Combinatorial Proof of the Non-Vanishing of Hankel Determinants of the Thue-Morse Sequence2014-08-21T10:24:22+10:00Yann Bugeaudbugeaud@math.unistra.frGuo-Niu Hanguoniu.han@unistra.fr<p>In 1998, Allouche, Peyrière, Wen and Wen established that the Hankel determinants associated with the Thue-Morse sequence on $\{-1,1\}$ are always nonzero. Their proof depends on a set of sixteen recurrence relations. We present an alternative, purely combinatorial proof of the same result. We also re-prove a recent result of Coons on the non-vanishing of the Hankel determinants associated to two other classical integer sequences.</p>2014-08-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p27Note on the Subgraph Component Polynomial2014-08-21T10:24:22+10:00Yunhua Liao307156168@qq.comYaoping Houyphou@hunnu.edu.cn<p>Tittmann, Averbouch and Makowsky [The enumeration of vertex induced subgraphs with respect to the number of components, <em>European J. Combin.</em> 32 (2011) 954-974] introduced the subgraph component polynomial $Q(G;x,y)$ of a graph $G$, which counts the number of connected components in vertex induced subgraphs. This polynomial encodes a large amount of combinatorial information about the underlying graph, such as the order, the size, and the independence number. We show that several other graph invariants, such as the connectivity and the number of cycles of length four in a regular bipartite graph are also determined by the subgraph component polynomial. Then, we prove that several well-known families of graphs are determined by the polynomial $Q(G;x,y).$ Moreover, we study the distinguishing power and find simple graphs which are not distinguished by the subgraph component polynomial but distinguished by the characteristic polynomial, the matching polynomial and the Tutte polynomial. These are partial answers to three open problems proposed by Tittmann et al.</p>2014-08-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p28Resolving a Conjecture on Degree of Regularity of Linear Homogeneous Equations2014-08-21T10:24:22+10:00Noah Golowichnoah_g@verizon.netA linear equation is $r$-regular, if, for every $r$-coloring of the positive integers, there exist positive integers of the same color which satisfy the equation. In 2005, Fox and Radoićič conjectured that the equation $x_1 + 2x_2 + \cdots + 2^{n-2}x_{n-1} - 2^{n-1}x_n = 0$, for any $n \geq 2$, has a degree of regularity of $n-1$, which would verify a conjecture of Rado from 1933. Rado's conjecture has since been verified with a different family of equations. In this paper, we show that Fox and Radoićič's family of equations indeed have a degree of regularity of $n-1$. We also prove a few extensions of this result.<br /><br />2014-08-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p29Additional Constructions to Solve the Generalized Russian Cards Problem using Combinatorial Designs2014-08-21T10:24:23+10:00Colleen M. Swansoncmswnsn@umich.eduDouglas R. Stinsondstinson@uwaterloo.ca<p>In the generalized Russian cards problem, we have a card deck $X$ of $n$ cards and three participants, Alice, Bob, and Cathy, dealt $a$, $b$, and $c$ cards, respectively. Once the cards are dealt, Alice and Bob wish to privately communicate their hands to each other via public announcements, without the advantage of a shared secret or public key infrastructure. Cathy, for her part, should remain ignorant of all but her own cards after Alice and Bob have made their announcements. Notions for Cathy's ignorance in the literature range from Cathy not learning the fate of any individual card with certainty (<em>weak $1$-security</em>) to not gaining any probabilistic advantage in guessing the fate of some set of $\delta$ cards (<em>perfect $\delta$-security</em>). As we demonstrate in this work, the generalized Russian cards problem has close ties to the field of combinatorial designs, on which we rely heavily, particularly for perfect security notions. Our main result establishes an equivalence between perfectly $\delta$-secure strategies and $(c+\delta)$-designs on $n$ points with block size $a$, when announcements are chosen uniformly at random from the set of possible announcements. We also provide construction methods and example solutions, including a construction that yields perfect $1$-security against Cathy when $c=2$. Drawing on our equivalence results, we are able to use a known combinatorial design to construct a strategy with $a=8$, $b=13$, and $c=3$ that is perfectly $2$-secure. Finally, we consider a variant of the problem that yields solutions that are easy to construct and optimal with respect to both the number of announcements and level of security achieved. Moreover, this is the first method obtaining weak $\delta$-security that allows Alice to hold an arbitrary number of cards and Cathy to hold a set of $c = \lfloor \frac{a-\delta}{2} \rfloor$ cards. Alternatively, the construction yields solutions for arbitrary $\delta$, $c$ and any $a \geq \delta + 2c$.</p>2014-08-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p30Capturing the Drunk Robber on a Graph2014-08-21T10:24:23+10:00Natasha Komarovnatalie.komarov@dartmouth.eduPeter Winklerpeter.winkler@dartmouth.edu<p><!--StartFragment-->We show that the expected time for a smart "cop"' to catch a drunk "robber" on an $n$-vertex graph is at most $n + {\rm o}(n)$. More precisely, let $G$ be a simple, connected, undirected graph with distinguished points $u$ and $v$ among its $n$ vertices. A cop begins at $u$ and a robber at $v$; they move alternately from vertex to adjacent vertex. The robber moves randomly, according to a simple random walk on $G$; the cop sees all and moves as she wishes, with the object of "capturing" the robber<span>—</span>that is, occupying the same vertex—in least expected time. We show that the cop succeeds in expected time no more than $n {+} {\rm o}(n)$. Since there are graphs in which capture time is at least $n {-} o(n)$, this is roughly best possible. We note also that no function of the diameter can be a bound on capture time.</p>2014-08-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p31On Compact Symmetric Regularizations of Graphs2014-08-21T10:24:23+10:00R. Vandellvandellr@ipfw.eduM. Walshwalshm@ipfw.eduW. D. Weakleyweakley@ipfw.edu<p>Let $G$ be a finite simple graph of order $n$, maximum degree $\Delta$, and minimum degree $\delta$. A <em>compact regularization</em> of $G$ is a $\Delta$-regular graph $H$ of which $G$ is an induced subgraph: $H$ is <em>symmetric</em> if every automorphism of $G$ can be extended to an automorphism of $H$. The <em>index</em> $|H:G|$ of a regularization $H$ of $G$ is the ratio $|V(H)|/|V(G)|$. Let $\mbox{mcr}(G)$ denote the index of a minimum compact regularization of $G$ and let $\mbox{mcsr}(G)$ denote the index of a minimum compact symmetric regularization of $G$.</p><p>Erdős and Kelly proved that every graph $G$ has a compact regularization and $\mbox{mcr}(G) \leq 2$. Building on a result of <span>König</span>, Chartrand and Lesniak showed that every graph has a compact symmetric regularization and $\mbox{mcsr}(G) \leq 2^{\Delta - \delta}$. Using a partial Cartesian product construction, we improve this to $\mbox{mcsr}(G) \leq \Delta - \delta + 2$ and give examples to show this bound cannot be reduced below $\Delta - \delta + 1$.</p>2014-08-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p32On Symmetry of Uniform and Preferential Attachment Graphs2014-08-28T09:50:25+10:00Abram Magneranmagner@purdue.eduSvante Jansonsvante@math.uu.seGiorgos Kolliasgkollias@us.ibm.comWojciech Szpankowskispa@cs.purdue.edu<p>Motivated by the problem of graph structure compression under realistic source models, we study the symmetry behavior of preferential and uniform attachment graphs. These are two dynamic models of network growth in which new nodes attach to a constant number $m$ of existing ones according to some attachment scheme. We prove symmetry results for $m=1$ and $2$, and we conjecture that for $m\geq 3$, both models yield asymmetry with high probability. We provide new empirical evidence in terms of graph defect. We also prove that vertex defects in the uniform attachment model grow at most logarithmically with graph size, then use this to prove a weak asymmetry result for all values of $m$ in the uniform attachment model. Finally, we introduce a natural variation of the two models that incorporates preference of new nodes for nodes of a similar age, and we show that the change introduces symmetry for all values of $m$.</p>2014-08-28T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p33A $q$-Queens Problem. I. General Theory2014-08-28T09:50:43+10:00Seth Chaikensdc@cs.albany.eduChristopher R. H. Hanusachanusa@qc.cuny.eduThomas Zaslavskyzaslav@math.binghamton.edu<p>By means of the Ehrhart theory of inside-out polytopes we establish a general counting theory for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen, on a polygonal convex board. The number of ways to place $q$ identical nonattacking pieces on a board of variable size $n$ but fixed shape is (up to a normalization) given by a quasipolynomial function of $n$, of degree $2q$, whose coefficients are polynomials in $q$. The number of combinatorially distinct types of nonattacking configuration is the evaluation of our quasipolynomial at $n=-1$. The quasipolynomial has an exact formula that depends on a matroid of weighted graphs, which is in turn determined by incidence properties of lines in the real affine plane. We study the highest-degree coefficients and also the period of the quasipolynomial, which is needed if the quasipolynomial is to be interpolated from data, and which is bounded by some function, not well understood, of the board and the piece's move directions.</p>2014-08-28T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p34On the Typical Structure of Graphs in a Monotone Property2014-08-28T09:50:56+10:00Svante Jansonsvante.janson@math.uu.seAndrew J. Uzzellandrew.uzzell@math.uu.seGiven a graph property $\mathcal{P}$, it is interesting to determine the typical structure of graphs that satisfy $\mathcal{P}$. In this paper, we consider monotone properties, that is, properties that are closed under taking subgraphs. Using results from the theory of graph limits, we show that if $\mathcal{P}$ is a monotone property and $r$ is the largest integer for which every $r$-colorable graph satisfies $\mathcal{P}$, then almost every graph with $\mathcal{P}$ is close to being a balanced $r$-partite graph.2014-08-28T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p35Arithmetic Properties of Overcubic Partition Pairs2014-09-04T12:43:43+10:00Bernard L.S. Linlinlsjmu@163.com<p>Let $\overline{b}(n)$ denote the number of overcubic partition pairs of $n$. In this paper, we establish two Ramanujan type congruences and several infinite families of congruences modulo $3$ satisfied by $\overline{b}(n)$ . For modulus $5$, we obtain one Ramanujan type congruence and two congruence relations for $\overline{b}(n)$, from which some strange congruences are derived.</p>2014-09-04T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p36Orthogonality and Minimality in the Homology of Locally Finite Graphs2014-09-11T06:03:58+10:00Reinhard DiestelReinhardDiestel@math.uni-hamburg.deJulian Pottjulianpott@gmail.com<p>Given a finite set $E$, a subset $D\subseteq E$ (viewed as a function $E\to \mathbb F_2$) is orthogonal to a given subspace $\mathcal F$ of the $\mathbb F_2$-vector space of functions $E\to \mathbb F_2$ as soon as $D$ is orthogonal to every $\subseteq$-minimal element of $\mathcal F$. This fails in general when $E$ is infinite.</p><p>However, we prove the above statement for the six subspaces $\mathcal F$ of the edge space of any $3$-connected locally finite graph that are relevant to its homology: the topological, algebraic, and finite cycle and cut spaces. This solves a problem of Diestel (2010, arXiv:0912.4213).</p>2014-09-11T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p37The Generating Function for Total Displacement2014-09-11T06:04:35+10:00Mathieu Guay-Paquetmathieu.guaypaquet@lacim.caKyle Petersentpeter21@depaul.eduIn a 1977 paper, Diaconis and Graham studied what Knuth calls the total displacement of a permutation $w$, which is the sum of the distances $|w(i)-i|$. In recent work of the first author and Tenner, this statistic appears as twice the type $A_{n-1}$ version of a statistic for Coxeter groups called the depth of $w$. There are various enumerative results for this statistic in the work of Diaconis and Graham, codified as exercises in Knuth's textbook, and some other results in the work of Petersen and Tenner. However, no formula for the generating function of this statistic appears in the literature. Knuth comments that "the generating function for total displacement does not appear to have a simple form." In this paper, we translate the problem of computing the distribution of total displacement into a problem of counting weighted Motzkin paths. In this way, standard techniques allow us to express the generating function for total displacement as a continued fraction.2014-09-11T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p38A Laurent Series Proof of the Habsieger-Kadell $q$-Morris Identity2014-09-18T10:44:05+10:00Xin Guoceguoce.xin@gmail.comZhou Yuenkzhouyue@gmail.com<p>We give a Laurent series proof of the Habsieger-Kadell $q$-Morris identity, which is a common generalization of the $q$-Morris identity and the Aomoto constant term identity. The proof allows us to extend the theorem for some additional parameter cases.</p>2014-09-18T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p39Distance-Restricted Matching Extension in Triangulations of the Torus and the Klein Bottle2014-09-18T10:44:21+10:00Robert E.L. Aldredraldred@maths.otago.ac.nzJun Fujisawafujisawa@fbc.keio.ac.jp<p>A graph $G$ with at least $2m+2$ edges is said to be distance $d$ $m$-extendable if for any matching $M$ in $G$ with $m$ edges in which the edges lie pair-wise distance at least $d$, there exists a perfect matching in $G$ containing $M$. In a previous paper, Aldred and Plummer proved that every $5$-connected triangulation of the plane or the projective plane of even order is distance $5$ $m$-extendable for any $m$. In this paper we prove that the same conclusion holds for every triangulation of the torus or the Klein bottle.</p>2014-09-18T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p40Bounding the Distinguishing Number of Infinite Graphs and Permutation Groups2014-09-18T10:44:33+10:00Simon M. Smithsismith@citytech.cuny.eduMark E. Watkinsmewatkin@syr.edu<p>A group of permutations $G$ of a set $V$ is $k$-<em>distinguishable</em> if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its <em>distinguishing number</em> $D(G,V)$. In particular, a graph $\Gamma$ is $k$-<em>distinguishable</em> if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.</p><p>Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is not $k$-distinguishable but in which every ball of finite radius is $k$-distinguishable.</p><p>In the second half of this paper we show that a large distinguishing number for an imprimitive group $G$ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action by $G$ on the corresponding system of imprimitivity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.</p>2014-09-18T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p41A Schur-Like Basis of NSym Defined by a Pieri Rule2014-09-18T10:44:48+10:00John Maxwell Campbellmaxwell8@yorku.caKaren Feldmanchagit@yorku.caJennifer Lightlightj@yorku.caPavel Shuldinerpavelshu@yorku.caYan Xusimonxiu@yorku.caRecent research on the algebra of non-commutative symmetric functions and the dual algebra of quasi-symmetric functions has explored some natural analogues of the Schur basis of the algebra of symmetric functions. We introduce a new basis of the algebra of non-commutative symmetric functions using a right Pieri rule. The commutative image of an element of this basis indexed by a partition equals the element of the Schur basis indexed by the same partition and the commutative image is $0$ otherwise. We establish a rule for right-multiplying an arbitrary element of this basis by an arbitrary element of the ribbon basis, and a Murnaghan-Nakayama-like rule for this new basis. Elements of this new basis indexed by compositions of the form $(1^n, m, 1^r)$ are evaluated in terms of the complete homogeneous basis and the elementary basis.2014-09-18T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p42A Polynomial Invariant and Duality for Triangulations2014-09-18T10:45:06+10:00Vyacheslav Krushkalkrushkal@virginia.eduDavid Renardydrenardy@umich.edu<p>The Tutte polynomial ${T}_G(X,Y)$ of a graph $G$ is a classical invariant, important in combinatorics and <span style="font-size: 10px;">statistical mechanics. An essential feature of the Tutte polynomial is the duality for planar graphs $G$, </span><span style="font-size: 10px;">$T_G(X,Y) = {T}_{G^*}(Y,X)$ where $G^*$ denotes the dual graph. We examine this property from the perspective </span><span style="font-size: 10px;">of manifold topology, formulating polynomial invariants for higher-dimensional simplicial complexes. </span><span style="font-size: 10px;">Polynomial duality for triangulations of a sphere follows as a consequence of Alexander duality. </span></p><p>The main goal of this paper is to introduce and begin the study of a more general $4$-variable polynomial for triangulations and handle decompositions of orientable manifolds. Polynomial duality in this case is a consequence of <span>Poincaré</span> duality on manifolds. In dimension 2 these invariants specialize to the well-known polynomial invariants of ribbon graphs defined by B. <span>Bollobás</span> and O. Riordan. Examples and specific evaluations of the polynomials are discussed.</p>2014-09-18T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p43Proof of the List Edge Coloring Conjecture for Complete Graphs of Prime Degree2014-09-18T10:45:18+10:00Uwe Schauzuwe.schauz@gmx.de<br />We prove that the list-chromatic index and paintability index of $K_{p+1}$ is $p$, for all odd primes $p$. This implies that the List Edge Coloring Conjecture holds for complete graphs with less then 10 vertices. It also shows that there are arbitrarily big complete graphs for which the conjecture holds, even among the complete graphs of class 1. Our proof combines the Quantitative Combinatorial Nullstellensatz with the Paintability Nullstellensatz and a group action on symmetric Latin squares. It displays various ways of using different Nullstellensätze. We also obtain a partial proof of a version of Alon and Tarsi's Conjecture about even and odd Latin squares.2014-09-18T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p44Energies and Structure of Additive Sets2014-09-18T10:45:34+10:00Shkredov Ilyailya.shkredov@gmail.com<p>In this paper we prove that any sumset or difference set has large $\textsf{E}_3$ energy. Also, we give a full description of families of sets having critical relations between some kind of energies such as $\textsf{E}_k$, $\textsf{T}_k$ and Gowers norms. In particular, we give criteria for a set to be a </p><ul><li>set of the form $H\dotplus \Lambda$, where $H+H$ is small and $\Lambda$ has "random structure",</li><li>set equal to a disjoint union of sets $H_j$ each with small doubling,</li><li>set having a large subset $A'$ with $2A'$ equal to a set with small doubling and $|A'+A'| \approx |A|^4 / \textsf{E}(A)$.</li></ul>2014-09-18T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p45A Note on a Ramsey-Type Problem for Sequences2014-09-18T10:45:48+10:00Andrzej Dudekandrzej.dudek@wmich.edu<p>Two sequences $\{x_i\}_{i=1}^{t}$ and $\{y_i\}_{i=1}^t$ of distinct integers are <em>similar</em> if their entries are order-isomorphic. Let $f(r,X)$ be the length of the shortest sequence $Y$ such that any $r$-coloring of the entries of $Y$ yields a monochromatic subsequence that is also similar to $X$. In this note we show that for any fixed non-monotone sequence $X$, $f(r,X)=\Theta(r^2)$, otherwise, for a monotone $X$, $f(r,X)=\Theta(r)$.</p>2014-09-18T00:00:00+10:00