http://www.combinatorics.org/ojs/index.php/eljc/issue/feedThe Electronic Journal of Combinatorics2016-02-05T12:16:18+11:00Matthias Beckbeck@math.sfsu.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/v23i1p1Bi-Cohen-Macaulay graphs2016-01-11T13:05:27+11:00Jürgen Herzogjuergen.herzog@uni-essen.deAhad Rahimiahad.rahimi@razi.ac.irIn this paper we consider bi-Cohen-Macaulay graphs, and give a complete classification of such graphs in the case they are bipartite or chordal. General bi-Cohen-Macaulay graphs are classified up to separation. The inseparable bi-Cohen-Macaulay graphs are determined. We establish a bijection between the set of all trees and the set of inseparable bi-Cohen-Macaulay graphs.<br /><br />2016-01-11T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p2On Periodicity of Generalized Pseudostandard Words2016-01-11T13:05:27+11:00Josef Florianflorijos@fjfi.cvut.czL'ubomíra Dvořákoválubomira.balkova@gmail.com<p>Generalized pseudostandard words were introduced by de Luca and De Luca in 2006. In comparison to the palindromic and pseudopalindromic closure, only little is known about the generalized pseudopalindromic closure and the associated generalized pseudostandard words. In this paper we provide a necessary and sufficient condition for their periodicity over a binary and a ternary alphabet. More precisely, we describe how the directive bi-sequence of a generalized pseudostandard word has to look like in order to correspond to a periodic word. We state moreover a conjecture concerning a necessary and sufficient condition for periodicity over any alphabet.</p>2016-01-11T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p3Posets of Finite Functions2016-01-11T13:05:27+11:00Konrad Piórokpioro@mimuw.edu.plThe symmetric group $ S(n) $ is partially ordered by Bruhat order. This order is extended by L. Renner to the set of partial injective functions of $ \{ 1, 2, \ldots, n \} $ (see, <em>Linear Algebraic Monoids</em>, Springer, 2005). This poset is investigated by M. Fortin in his paper <em>The MacNeille Completion of the Poset of Partial Injective Functions</em> [Electron. J. Combin., 15, R62, 2008]. In this paper we show that Renner order can be also defined for sets of all functions, partial functions, injective and partial injective functions from $ \{ 1, 2, \ldots, n \} $ to $ \{ 1, 2, \ldots, m \} $. Next, we generalize Fortin's results on these posets, and also, using simple facts and methods of linear algebra, we give simpler and shorter proofs of some fundamental Fortin's results. We first show that these four posets can be order embedded in the set of $ n \times m $-matrices with non-negative integer entries and with the natural componentwise order. Second, matrix representations of the Dedekind-MacNeille completions of our posets are given. Third, we find join- and meet-irreducible elements for every finite sublattice of the lattice of all $ n \times m $-matrices with integer entries. In particular, we obtain join- and meet-irreducible elements of these Dedekind-MacNeille completions. Hence and by general results concerning Dedekind-MacNeille completions, join- and meet-irreducible elements of our four posets of functions are also found. Moreover, subposets induced by these irreducible elements are precisely described.2016-01-11T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p4Simultaneous Core Partitions: Parameterizations and Sums2016-01-11T13:05:27+11:00Victor Y. Wangvywang@mit.edu<p>Fix coprime $s,t\ge1$. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the finitely many simultaneous $(s,t)$-cores have average size $\frac{1}{24}(s-1)(t-1)(s+t+1)$, and that the subset of self-conjugate cores has the same average (first shown by Chen-Huang-Wang). We similarly prove a recent conjecture of Fayers that the average weighted by an inverse stabilizer - giving the "expected size of the $t$-core of a random $s$-core" - is $\frac{1}{24}(s-1)(t^2-1)$. We also prove Fayers' conjecture that the analogous self-conjugate average is the same if $t$ is odd, but instead $\frac{1}{24}(s-1)(t^2+2)$ if $t$ is even. In principle, our explicit methods - or implicit variants thereof - extend to averages of arbitrary powers.</p><p>The main new observation is that the stabilizers appearing in Fayers' conjectures have simple formulas in Johnson's $z$-coordinates parameterization of $(s,t)$-cores.</p><p>We also observe that the $z$-coordinates extend to parameterize general $t$-cores. As an example application with $t := s+d$, we count the number of $(s,s+d,s+2d)$-cores for coprime $s,d\ge1$, verifying a recent conjecture of Amdeberhan and Leven.</p><div id="spoon-plugin-kncgbdglledmjmpnikebkagnchfdehbm-2" style="display: none;"> </div>2016-01-11T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p5An Extension of MacMahon's Equidistribution Theorem to Ordered Multiset Partitions2016-01-11T13:05:27+11:00Andrew Timothy Wilsonandwils@math.upenn.eduA classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work, we prove a strengthening of MacMahon's theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. This generalization leads to a new extension of Macdonald polynomials for hook shapes. We use our main theorem to show that these polynomials are symmetric and we give their Schur expansion.2016-01-11T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p6Low Degree Nullstellensatz Certificates for 3-Colorability2016-01-11T13:05:27+11:00Bo Libli@g.hmc.eduBenjamin Lowensteinblowenstein@g.hmc.eduMohamed Omaromar@g.hmc.eduIn a seminal paper, De Loera et. al introduce the algorithm NulLA (Nullstellensatz Linear Algebra) and use it to measure the difficulty of determining if a graph is not 3-colorable. The crux of this relies on a correspondence between 3-colorings of a graph and solutions to a certain system of polynomial equations over a field $\mathbb{k}$. In this article, we give a new direct combinatorial characterization of graphs that can be determined to be non-3-colorable in the first iteration of this algorithm when $\mathbb{k}=GF(2)$. This greatly simplifies the work of De Loera et. al, as we express the combinatorial characterization directly in terms of the graphs themselves without introducing superfluous directed graphs. Furthermore, for all graphs on at most $12$ vertices, we determine at which iteration NulLA detects a graph is not 3-colorable when $\mathbb{k}=GF(2)$.2016-01-11T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p7Expansions of a Chord Diagram and Alternating Permutations2016-01-11T13:05:27+11:00Tomoki Nakamigawanakami@info.shonan-it.ac.jpA chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram $E$ with $n$ chords is called an $n$-crossing if all chords of $E$ are mutually crossing. A chord diagram $E$ is called nonintersecting if $E$ contains no $2$-crossing. For a chord diagram $E$ having a $2$-crossing $S = \{ x_1 x_3, x_2 x_4 \}$, the expansion of $E$ with respect to $S$ is to replace $E$ with $E_1 = (E \setminus S) \cup \{ x_2 x_3, x_4 x_1 \}$ or $E_2 = (E \setminus S) \cup \{ x_1 x_2, x_3 x_4 \}$. It is shown that there is a one-to-one correspondence between the multiset of all nonintersecting chord diagrams generated from an $n$-crossing with a finite sequence of expansions and the set of alternating permutations of order $n+1$.2016-01-11T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p8Combinatorial Proofs of Addition Formulas2016-01-11T13:05:27+11:00Xiang-Ke Changchangxk@lsec.cc.ac.cnXing-Biao Huhxb@lsec.cc.ac.cnHongchuan Leihongchuanlei@gmail.comYeong-Nan Yehmayeh@math.sinica.edu.tw<p><!--StartFragment-->In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the $LDU$ decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition formula for weighted large Schröder paths.<!--EndFragment--></p>2016-01-11T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p9Sphere Representations, Stacked Polytopes, and the Colin de Verdière Number of a Graph2016-01-22T12:19:18+11:00Lon Mitchelllhm@ams.orgLynne Yengulalplyengulalp1@udayton.eduWe prove that a $k$-tree can be viewed as a subgraph of a special type of $(k+1)$-tree that corresponds to a stacked polytope and that these "stacked'' $(k+1)$-trees admit representations by orthogonal spheres in $\mathbb{R}^{k+1}$. As a result, we derive lower bounds for Colin de Verdière's $\mu$ of complements of partial $k$-trees and prove that $\mu(G) + \mu(\overline{G}) \geq |G| - 2$ for all chordal $G$.2016-01-22T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p10Finite Edge-Transitive Oriented Graphs of Valency Four: a Global Approach2016-01-22T12:20:33+11:00Jehan A. Al-barjalbar@kau.edu.saAhmad N. Al-kenanianalkenani@kau.edu.saNajat M. Muthananmuthana@kau.edu.saCheryl E. Praegercheryl.praeger@uwa.edu.auPablo Spigapablo.spiga@unimib.itWe develop a new framework for analysing finite connected, oriented graphs of valency four, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of `basic' graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restrictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.2016-01-22T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p11A Quantitative Study of Pure Parallel Processes2016-01-22T12:22:13+11:00O. BodiniOlivier.Bodini@lipn.univ-paris13.frA. Genitriniantoine.genitrini@lip6.frF. Peschanskifrederic.peschanski@lip6.frIn this paper, we study the interleaving <em></em><em>–</em> or pure merge <em></em><em></em><em>–</em> operator that most often characterizes parallelism in concurrency theory. This operator is a principal cause of the so-called combinatorial explosion that makes the analysis of process behaviours e.g. by model-checking, very hard <em></em><em>–</em><em></em> at least from the point of view of computational complexity. The originality of our approach is to study this combinatorial explosion phenomenon on average, relying on advanced analytic combinatorics techniques. We study various measures that contribute to a better understanding of the process behaviours represented as plane rooted trees: the number of runs (corresponding to the width of the trees), the expected total size of the trees as well as their overall shape. Two practical outcomes of our quantitative study are also presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random sampling of concurrent runs. These provide interesting responses to the combinatorial explosion problem.2016-01-22T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p12The Phase Transition in Site Percolation on Pseudo-Random Graphs2016-01-22T12:23:14+11:00Michael Krivelevichkrivelev@post.tau.ac.ilWe establish the existence of the phase transition in site percolation on pseudo-random $d$-regular graphs. Let $G=(V,E)$ be an $(n,d,\lambda)$-graph, that is, a $d$-regular graph on $n$ vertices in which all eigenvalues of the adjacency matrix, but the first one, are at most $\lambda$ in their absolute values. Form a random subset $R$ of $V$ by putting every vertex $v\in V$ into $R$ independently with probability $p$. Then for any small enough constant $\epsilon>0$, if $p=\frac{1-\epsilon}{d}$, then with high probability all connected components of the subgraph of $G$ induced by $R$ are of size at most logarithmic in $n$, while for $p=\frac{1+\epsilon}{d}$, if the eigenvalue ratio $\lambda/d$ is small enough as a function of $\epsilon$, then typically $R$ contains a connected component of size at least $\frac{\epsilon n}{d}$ and a path of length proportional to $\frac{\epsilon^2n}{d}$.2016-01-22T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p13A Combinatorial Proof of a Relationship Between Maximal $(2k-1,2k+1)$-Cores and $(2k-1,2k,2k+1)$-Cores2016-01-22T12:24:26+11:00Rishi Nathrnath@york.cuny.eduJames A. Sellerssellersj@psu.eduInteger partitions which are simultaneously $t$-cores for distinct values of $t$ have attracted significant interest in recent years. When $s$ and $t$ are relatively prime, Olsson and Stanton have determined the size of the maximal $(s,t)$-core $\kappa_{s,t}$. When $k\geq 2$, a conjecture of Amdeberhan on the maximal $(2k-1,2k,2k+1)$-core $\kappa_{2k-1,2k,2k+1}$ has also recently been verified by numerous authors.<br /><br />In this work, we analyze the relationship between maximal $(2k-1,2k+1)$-cores and maximal $(2k-1,2k,2k+1)$-cores. In previous work, the first author noted that, for all $k\geq 1,$<br />$$<br />\vert \, \kappa_{2k-1,2k+1}\, \vert = 4\vert \, \kappa_{2k-1,2k,2k+1}\, \vert<br />$$<br />and requested a combinatorial interpretation of this unexpected identity. Here, using the theory of abaci, partition dissection, and elementary results relating triangular numbers and squares, we provide such a combinatorial proof.2016-01-22T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p14The Chromatic Number of a Signed Graph2016-01-22T12:26:02+11:00Edita Máčajovámacajova@dcs.fmph.uniba.skAndré Raspaudraspaud@labri.frMartin Škovieraskoviera@dcs.fmph.uniba.sk<p>In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $\phi\colon V(G)\to \mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $\phi(u)$ is different from the colour $\sigma(uv)\phi(v)$, where is $\sigma(uv)$ is the sign of the edge $uv$. The substantial part of Zaslavsky's research concentrated on polynomial invariants related to signed graph colourings rather than on the behaviour of colourings of individual signed graphs. We continue the study of signed graph colourings by proposing the definition of a chromatic number for signed graphs which provides a natural extension of the chromatic number of an unsigned graph. We establish the basic properties of this invariant, provide bounds in terms of the chromatic number of the underlying unsigned graph, investigate the chromatic number of signed planar graphs, and prove an extension of the celebrated Brooks' theorem to signed graphs.</p>2016-01-22T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p15Chromatic Bases for Symmetric Functions2016-01-22T12:27:08+11:00Soojin Chochosj@ajou.ac.krStephanie van Willigenburgsteph@math.ubc.caIn this note we obtain numerous new bases for the algebra of symmetric functions whose generators are chromatic symmetric functions. More precisely, if $\{ G_ k \} _{k\geq 1}$ is a set of connected graphs such that $G_k$ has $k$ vertices for each $k$, then the set of all chromatic symmetric functions $\{ X_{G_ k} \} _{k\geq 1}$ generates the algebra of symmetric functions. We also obtain explicit expressions for the generators arising from complete graphs, star graphs, path graphs and cycle graphs.2016-01-22T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p16A Note on Perfect Matchings in Uniform Hypergraphs2016-01-22T12:28:08+11:00Andrew Treglowna.c.treglown@bham.ac.ukYi Zhaoyzhao6@gsu.eduWe determine the <em>exact</em> minimum $\ell$-degree threshold for perfect matchings in $k$-uniform hypergraphs when the corresponding threshold for perfect fractional matchings is significantly less than $\frac{1}{2}\left( \begin{array}{c} n \\ k- \ell\end{array}\right)$. This extends our previous results that determine the minimum $\ell$-degree thresholds for perfect matchings in $k$-uniform hypergraphs for all $\ell\ge k/2$ and provides two new (exact) thresholds: $(k,\ell)=(5,2)$ and $(7,3)$.<br /><br /><br />2016-01-22T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p17A Note on Maxima in Random Walks2016-01-22T12:29:03+11:00Joseph Helferjoseph.helfer@mail.mcgill.caDaniel T. Wisedaniel.wise@mcgill.ca<p>We give a combinatorial proof that a random walk attains a unique maximum with probability at least $1/2$. For closed random walks with uniform step size, we recover Dwass's count of the number of length $\ell$ walks attaining the maximum exactly $k$ times. We also show that the probability that there is both a unique maximum and a unique minimum is asymptotically equal to $\frac14$ and that the probability that a Dyck word has a unique minimum is asymptotically $\frac12$.</p>2016-01-22T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p18Avoiding Letter Patterns in Ternary Square-Free Words2016-02-05T12:13:34+11:00Elena A. Petrovaje.a.petrova@gmail.comWe consider special patterns of lengths 5 and 6 in a ternary alphabet. We show that some of them are unavoidable in square-free words and prove avoidability of the other ones. Proving the main results, we use Fibonacci words as codes of ternary words in some natural coding system and show that they can be decoded to square-free words avoiding the required patterns. Furthermore, we estimate the minimal local (critical) exponents of square-free words with such avoidance properties.2016-02-05T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p19Doubled Patterns are 3-Avoidable2016-02-05T12:14:01+11:00Pascal Ochemochem@lirmm.frIn combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $f$ of $w$ such that $f=h(p)$ where $h: \Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. A pattern is said to be doubled if no variable occurs only once. Doubled patterns with at most 3 variables and doubled patterns with at least 6 variables are $3$-avoidable. We show that doubled patterns with 4 and 5 variables are also $3$-avoidable.<br /><br />2016-02-05T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p20A Note on the $\gamma$-Coefficients of the Tree Eulerian Polynomial2016-02-05T12:14:20+11:00Rafael S. González D'Leónrafaeldleon@gmail.com<p>We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. A classical product formula shows that this polynomial factors completely over the integers. From this product formula it can be concluded that this polynomial has positive coefficients in the $\gamma$-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from previous results of Liu, Dotsenko-Khoroshkin, Bershtein-Dotsenko-Khoroshkin, González D'León-Wachs and Gonzláez D'León related to the free multibracketed Lie algebra and the poset of weighted partitions.</p>2016-02-05T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p21Another Proof of the Harer-Zagier Formula2016-02-05T12:14:36+11:00Boris Pittelbgp@math.ohio-state.eduFor a regular $2n$-gon there are $(2n-1)!!$ ways to match and glue the $2n$ sides. The Harer-Zagier bivariate generating function enumerates the gluings by $n$ and the genus $g$ of the attendant surface and leads to a recurrence equation for the counts of gluings with parameters $n$ and $g$. This formula was originally obtained using multidimensional Gaussian integrals. Soon after, Jackson and later Zagier found alternative proofs using symmetric group characters. In this note we give a different, characters-based, proof. Its core is computing and marginally inverting the Fourier transform of the underlying probability measure on $S_{2n}$. A key ingredient is the Murnaghan-Nakayama rule for the characters associated with one-hook Young diagrams.2016-02-05T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p22A New Lower Bound for the Towers of Hanoi Problem2016-02-05T12:14:53+11:00Codruƫ Grosugrosu.codrut@gmail.com<p>More than a century after its proposal, the Towers of Hanoi puzzle with 4 pegs was solved by Thierry Bousch in a breakthrough paper in 2014. The general problem with $p$ pegs is still open, with the best lower bound on the minimum number of moves due to Chen and Shen. We use some of Bousch's new ideas to obtain an asymptotic improvement on this bound for all $p \geq 5$.</p>2016-02-05T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p23An Orthogonal Basis for Functions over a Slice of the Boolean Hypercube2016-02-05T12:15:09+11:00Yuval Filmusfilmus.yuval@gmail.com<p>We present a simple, explicit orthogonal basis of eigenvectors for the Johnson and Kneser graphs, based on Young's orthogonal representation of the symmetric group. Our basis can also be viewed as an orthogonal basis for the vector space of all functions over a slice of the Boolean hypercube (a set of the form $\{(x_1,\ldots,x_n) \in \{0,1\}^n : \sum_i x_i = k\}$), which refines the eigenspaces of the Johnson association scheme; our basis is orthogonal with respect to <em>any</em> exchangeable measure. More concretely, our basis is an orthogonal basis for all multilinear polynomials $\mathbb{R}^n \to \mathbb{R}$ which are annihilated by the differential operator $\sum_i \partial/\partial x_i$. As an application of the last point of view, we show how to lift low-degree functions from a slice to the entire Boolean hypercube while maintaining properties such as expectation, variance and $L^2$-norm.</p><p><br />As an application of our basis, we streamline Wimmer's proof of Friedgut's theorem for the slice. Friedgut's theorem, a fundamental result in the analysis of Boolean functions, states that a Boolean function on the Boolean hypercube with low total influence can be approximated by a Boolean junta (a function depending on a small number of coordinates). Wimmer generalized this result to slices of the Boolean hypercube, working mostly over the symmetric group, and utilizing properties of Young's orthogonal representation. Using our basis, we show how the entire argument can be carried out directly on the slice.</p>2016-02-05T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p24Generalizing the Classic Greedy and Necklace Constructions of de Bruijn Sequences and Universal Cycles2016-02-05T12:15:27+11:00Joe Sawadajsawada@uoguelph.caAaron Williamsharon@uvic.caDennis Wongcwong@uoguelph.ca<p>We present a class of languages that have an interesting property: For each language $\mathbf{L}$ in the class, both the classic greedy algorithm and the classic Lyndon word (or necklace) concatenation algorithm provide the lexicographically smallest universal cycle for $\mathbf{L}$. The languages consist of length $n$ strings over $\{1,2,\ldots ,k\}$ that are closed under rotation with their subset of necklaces also being closed under replacing any suffix of length $i$ by $i$ copies of $k$. Examples include all strings (in which case universal cycles are commonly known as de Bruijn sequences), strings that sum to at least $s$, strings with at most $d$ cyclic descents for a fixed $d>0$, strings with at most $d$ cyclic decrements for a fixed $d>0$, and strings avoiding a given period. Our class is also closed under both union and intersection, and our results generalize results of several previous papers.</p>2016-02-05T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p25Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences2016-02-05T12:15:47+11:00Luke Schaefferlrschaeffer@gmail.comJeffrey Shallitshallit@cs.uwaterloo.ca<p>We prove that the property of being closed (resp., palindromic, rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of those $n$ for which an automatic sequence $\bf x$ has a closed (resp., palindromic, privileged, rich, trapezoidal, balanced) factor of length $n$ is itself automatic. For privileged words this requires a new characterization of the privileged property. We compute the corresponding characteristic functions for various famous sequences, such as the Thue-Morse sequence, the Rudin-Shapiro sequence, the ordinary paperfolding sequence, the period-doubling sequence, and the Fibonacci sequence. Finally, we also show that the function counting the total number of palindromic factors in the prefix of length $n$ of a $k$-automatic sequence is not $k$-synchronized.<br /><br /></p>2016-02-05T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p26Determining a Binary Matroid from its Small Circuits2016-02-05T12:16:04+11:00James Oxleyoxley@math.lsu.eduCharles Semplecharles.semple@canterbury.ac.nzGeoff Whittlegeoff.whittle@vuw.ac.nzIt is well known that a rank-$r$ matroid $M$ is uniquely determined by its circuits of size at most $r$. This paper proves that if $M$ is binary and $r\ge 3$, then $M$ is uniquely determined by its circuits of size at most $r-1$ unless $M$ is a binary spike or a special restriction thereof. In the exceptional cases, $M$ is determined up to isomorphism.2016-02-05T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p27A Curved Brunn Minkowski Inequality for the Symmetric Group2016-02-05T12:16:18+11:00Weerachai Neeranartvongweeracha@mit.eduJonathan Novakjnovak@math.mit.eduNat Sothanaphannsothana@mit.edu<p>In this paper, we construct an injection $A \times B \rightarrow M \times M$ from the product of any two nonempty subsets of the symmetric group into the square of their midpoint set, where the metric is that corresponding to the conjugacy class of transpositions. If $A$ and $B$ are disjoint, our construction allows to inject two copies of $A \times B$ into $M \times M$. These injections imply a positively curved Brunn-Minkowski inequality for the symmetric group analogous to that obtained by Ollivier and Villani for the hypercube. However, while Ollivier and Villani's inequality is optimal, we believe that the curvature term in our inequality can be improved. We identify a hypothetical concentration inequality in the symmetric group and prove that it yields an optimally curved Brunn-Minkowski inequality.<br /><br /></p>2016-02-05T00:00:00+11:00