Showing papers in "Electronic Journal of Combinatorics in 2006"
TL;DR: The computation of the multiplicities in this generalization of Faa di Bruno's formula is a combinatorial enumeration problem that, although completely elementary, seems to have been neglected.
Abstract: The natural forms of the Leibniz rule for the $k$th derivative of a product and of Faa di Bruno's formula for the $k$th derivative of a composition involve the differential operator $\partial^k/\partial x_1 \cdots \partial x_k$ rather than $d^k/dx^k$, with no assumptions about whether the variables $x_1,\dots,x_k$ are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables. Coefficients appearing in forms of these identities in which some variables are indistinguishable are just multiplicities of indistinguishable terms (in particular, if all variables are distinct then all coefficients are 1). The computation of the multiplicities in this generalization of Faa di Bruno's formula is a combinatorial enumeration problem that, although completely elementary, seems to have been neglected. We apply the results to cumulants of probability distributions.
169 citations
TL;DR: In this paper, the distinguishing chromatic number of a graph was defined and studied, and it was shown that the maximum chromatic distingushing number for trees can be computed for both trees and connected graphs.
Abstract: In this paper we define and study the distinguishing chromatic number, $\chi_D(G)$, of a graph $G$, building on the work of Albertson and Collins who studied the distinguishing number. We find $\chi_D(G)$ for various families of graphs and characterize those graphs with $\chi_D(G)$ $ = |V(G)|$, and those trees with the maximum chromatic distingushing number for trees. We prove analogs of Brooks' Theorem for both the distinguishing number and the distinguishing chromatic number, and for both trees and connected graphs. We conclude with some conjectures.
116 citations
102 citations
TL;DR: It is shown in this paper that the algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace – namely, the number of reaches of the directed graph.
Abstract: Let $G$ denote a directed graph with adjacency matrix $Q$ and in-degree matrix $D$. We consider the Kirchhoff matrix $L=D-Q$, sometimes referred to as the directed Laplacian . A classical result of Kirchhoff asserts that when $G$ is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of $G$. This fact has a meaningful generalization to directed graphs, as was recently observed by Chebotarev and Agaev in 2005. Since this result has many important applications in the sciences, we offer an independent and self-contained proof of their theorem, showing in this paper that the algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace – namely, the number of reaches of the directed graph. We also extend their results by deriving a natural basis for the corresponding eigenspace. The results are proved in the general context of stochastic matrices, and apply equally well to directed graphs with non-negative edge weights.
92 citations
TL;DR: The symmetric distribution of the numbers of crossings and nestings in partitions is derived, which generalizes a recent result of Klazar and Noy in perfect matchings by factorizing the involution through bijections between set partitions and some path diagrams.
Abstract: We construct an involution on set partitions which keeps track of the numbers of crossings, nestings and alignments of two edges We derive then the symmetric distribution of the numbers of crossings and nestings in partitions, which generalizes a recent result of Klazar and Noy in perfect matchings By factorizing our involution through bijections between set partitions and some path diagrams we obtain the continued fraction expansions of the corresponding ordinary generating functions
89 citations
TL;DR: In this paper, the authors use the line digraph construction to associate an orthogonal matrix with each graph and derive two further matrices from these matrices, each of which is considered as a graph invariant.
Abstract: We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the first two cases, we compute the spectrum explicitly and show that it is determined by the spectrum of the adjacency matrix of the original graph. We then show by computation that the isomorphism classes of many known families of strongly regular graphs (up to 64 vertices) are characterized by the spectrum of this matrix. We conjecture that this is always the case for strongly regular graphs and we show that the conjecture is not valid for general graphs. We verify that the smallest regular graphs which are not distinguished with our method are on 14 vertices.
84 citations
TL;DR: In this paper, it was shown that there exists a regular graph with geometric thickness at least (1 − ε,n − 1/2-4/ε-πσon) for all ε ≥ 0.
Abstract: The geometric thickness of a graph $G$ is the minimum integer $k$ such that there is a straight line drawing of $G$ with its edge set partitioned into $k$ plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs , vol. 342 of Contemp. Math. , AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists $\Delta$-regular graphs with arbitrarily large geometric thickness. In particular, for all $\Delta\geq9$ and for all large $n$, there exists a $\Delta$-regular graph with geometric thickness at least $c\sqrt{\Delta}\,n^{1/2-4/\Delta-\epsilon}$. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmovic et al. [Really straight graph drawings. In Proc. 12th International Symp. on Graph Drawing (GD '04), vol. 3383 of Lecture Notes in Comput. Sci. , Springer, 2004] and Ambrus et al. [The slope parameter of graphs. Tech. Rep. MAT-2005-07, Department of Mathematics, Technical University of Denmark, 2005].
81 citations
TL;DR: This paper describes ways of finding and verifying determining sets, gives natural lower bounds on the determining number, and shows how to use orbits to investigate determining sets.
Abstract: A set of vertices $S$ is a determining set for a graph $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The determining number of a graph is the size of a smallest determining set. This paper describes ways of finding and verifying determining sets, gives natural lower bounds on the determining number, and shows how to use orbits to investigate determining sets. Further, determining sets of Kneser graphs are extensively studied, sharp bounds for their determining numbers are provided, and all Kneser graphs with determining number $2$, $3,$ or $4$ are given.
81 citations
TL;DR: In this article, the authors introduced and characterised grid classes, which are natural generalisations of other well-studied permutation classes, and gave a new, short proof of the Fibonacci dichotomy: the number of permutations of length $n$ in a permutation class is either at least as large as the first or polynomial.
Abstract: We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of length $n$ in a permutation class is either at least as large as the $n$th Fibonacci number or is eventually polynomial
75 citations
TL;DR: The sizes of orbits from the action of PGL$(2,q) on the-subsets of the projective line when $k$ is not congruent to $0$ and 1 modulo $p$.
Abstract: The group PGL$(2,q)$, $q=p^n$, $p$ an odd prime, is $3$-transitive on the projective line and therefore it can be used to construct $3$-designs. In this paper, we determine the sizes of orbits from the action of PGL$(2,q)$ on the $k$-subsets of the projective line when $k$ is not congruent to $0$ and 1 modulo $p$. Consequently, we find all values of $\lambda$ for which there exist $3$-$(q+1,k,\lambda)$ designs admitting PGL$(2,q)$ as automorphism group. In the case $p\equiv 3$ mod 4, the results and some previously known facts are used to classify 3-designs from PSL$(2,p)$ up to isomorphism.
72 citations
TL;DR: The authors construct graphs with n vertices of maximum degree $5$ whose every straight-line drawing in the plane uses edges of at least $n^{1/6-o(1)}$ distinct slopes.
Abstract: We construct graphs with $n$ vertices of maximum degree $5$ whose every straight-line drawing in the plane uses edges of at least $n^{1/6-o(1)}$ distinct slopes.
TL;DR: A combinatorial derivation and interpretation of the weights associated with the stationary distribution of the partially asymmetric exclusion process are given and a direct proof of the validity of the weight equations is provided.
Abstract: We give a combinatorial derivation and interpretation of the weights associated with the stationary distribution of the partially asymmetric exclusion process. We define a set of weight equations, which the stationary distribution satisfies. These allow us to find explicit expressions for the stationary distribution and normalisation using both recurrences and path models. To show that the stationary distribution satisfies the weight equations, we construct a Markov chain on a larger set of generalised configurations. A bijection on this new set of configurations allows us to find the stationary distribution of the new chain. We then show that a subset of the generalised configurations is equivalent to the original process and that the stationary distribution on this subset is simply related to that of the original chain. We also provide a direct proof of the validity of the weight equations.
[...]
TL;DR: A new derivation of the threshold of appearance of the core of a random graph is given, using a hybrid model obtained from a simple model of random graphs based on random functions and the pairing or configuration model for random graphs with given degree sequence.
Abstract: We give a new derivation of the threshold of appearance of the $k$-core of a random graph. Our method uses a hybrid model obtained from a simple model of random graphs based on random functions, and the pairing or configuration model for random graphs with given degree sequence. Our approach also gives a simple derivation of properties of the degree sequence of the $k$-core of a random graph, in particular its relation to multinomial and hence independent Poisson variables. The method is also applied to $d$-uniform hypergraphs.
TL;DR: It is shown that the free dendriform trialgebra on one generator embeds in the free Baxter algebra with a quasi-idempotent map and an idempotent generator, which refines results of Ebrahimi-Fard and Guo.
Abstract: We study the free (associative, non-commutative) Baxter algebra on one generator. The first explicit description of this object is due to Ebrahimi-Fard and Guo. We provide an alternative description in terms of a certain class of trees, which form a linear basis for this algebra. We use this to treat other related cases, particularly that in which the Baxter map is required to be quasi-idempotent, in a unified manner. Each case corresponds to a different class of trees. Our main focus is on the underlying combinatorics. In several cases, we provide bijections between our various classes of trees and more familiar combinatorial objects including certain Schroder paths and Motzkin paths. We calculate the dimensions of the homogeneous components of these algebras (with respect to a bidegree related to the number of nodes and the number of angles in the trees) and the corresponding generating series. An important feature is that the combinatorics is captured by the idempotent case; the others are obtained from this case by various binomial transforms. We also relate free Baxter algebras to Loday's dendriform trialgebras and dialgebras. We show that the free dendriform trialgebra (respectively, dialgebra) on one generator embeds in the free Baxter algebra with a quasi-idempotent map (respectively, with a quasi-idempotent map and an idempotent generator). This refines results of Ebrahimi-Fard and Guo.
TL;DR: In this paper, Tamm showed that the generating function for Hankel determinants has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series.
Abstract: Michael Somos conjectured a relation between Hankel determinants whose entries ${1\over 2n+1}{3n\choose n}$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos's Hankel determinants to known determinants, and we obtain, up to a power of $3$, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of $r$-ary trees.
TL;DR: It is shown that there is a similar multiplicative relationship between the Ehrhart series for P, for Q, and for the free sum $P\oplus Q$ that holds when P is reflexive and Q contains $0$ in its interior.
Abstract: It is well known that for $P$ and $Q$ lattice polytopes, the Ehrhart polynomial of $P\times Q$ satisfies $L_{P\times Q}(t)=L_P(t)L_Q(t)$. We show that there is a similar multiplicative relationship between the Ehrhart series for $P$, for $Q$, and for the free sum $P\oplus Q$ that holds when $P$ is reflexive and $Q$ contains $0$ in its interior.
TL;DR: It is shown that the unreduced Euler characteristic of $\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten.
Abstract: Let $S_{m,n}$ be the graph on the vertex set ${\Bbb Z}_m \times {\Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $\Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $\det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to $\{\Sigma_{m,n} : n \ge 1\}$. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.
TL;DR: The motivation for creating tables of codes of lengths of lengths was their application to the generation of good sets of frequency hopping lists in radio networks.
Abstract: Existing tables of constant weight codes are mainly confined to codes of length $n\leq 28$ This paper presents tables of codes of lengths $29 \leq n \leq 63$ The motivation for creating these tables was their application to the generation of good sets of frequency hopping lists in radio networks The complete generation of all relevant cases by a small number of algorithms is augmented in individual cases by miscellaneous constructions These sometimes give a larger number of codewords than the algorithms
TL;DR: The approach gives a new proof of Du and Liu's results and gives new hook length formulas, and a interpretation of the polynomials P_n(a,b,c) in terms of parking functions: the authors count parking functions according to the number of cars that park in their preferred parking spaces.
Abstract: A proper vertex of a rooted tree with totally ordered vertices is a vertex that is the smallest of all its descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all expressed in terms of the polynomials
TL;DR: For undirected graphs, it has been known for some time that one can bound the diameter using the eigenvalues of the Laplacian as discussed by the authors, where the diameter of strongly connected directed graphs can be bounded using the Perron vector of the transition probability matrix.
Abstract: For undirected graphs it has been known for some time that one can bound the diameter using the eigenvalues In this note we give a similar result for the diameter of strongly connected directed graphs $G$, namely $$ D(G) \leq \bigg \lfloor {2\min_x \log (1/\phi(x))\over \log{2\over 2-\lambda}} \bigg\rfloor +1 $$ where $\lambda$ is the first non-trivial eigenvalue of the Laplacian and $\phi$ is the Perron vector of the transition probability matrix of a random walk on $G$
TL;DR: This paper presents O(n \log n)-time algorithms that compute the distinguishing numbers of trees and forests by relying on the combinatorial properties of trees rather than their automorphism groups to compute for their distinguishing numbers.
Abstract: Let $G$ be a graph. A vertex labeling of $G$ is distinguishing if the only label-preserving automorphism of $G$ is the identity map. The distinguishing number of $G$, $D(G)$, is the minimum number of labels needed so that $G$ has a distinguishing labeling. In this paper, we present $O(n \log n)$-time algorithms that compute the distinguishing numbers of trees and forests. Unlike most of the previous work in this area, our algorithm relies on the combinatorial properties of trees rather than their automorphism groups to compute for their distinguishing numbers.
TL;DR: In this paper, the factorial Grothendieck polynomials were studied from a combinatorial viewpoint and a generalisation of the Littlewood-Richardson rule was proposed.
Abstract: In this paper, we study Grothendieck polynomials indexed by Grassmannian permutations from a combinatorial viewpoint. We introduce the factorial Grothendieck polynomials which are analogues of the factorial Schur functions, study their properties, and use them to produce a generalisation of a Littlewood-Richardson rule for Grothendieck polynomials.
TL;DR: A simpler proof of the result of Boros and Furedi that for any finite set of points in the plane in general position there is a point lying in $2/9$ of all the triangles determined by these points.
Abstract: We give a simpler proof of the result of Boros and Furedi that for any finite set of points in the plane in general position there is a point lying in $2/9$ of all the triangles determined by these points.
TL;DR: The authors generalize the Ihara-Selberg zeta function to hypergraphs in a natural way, leading to exact factorization results for biregular bipartite graphs, and show that modified Riemann hypothesis is true if and only if the hypergraph is Ramanujan in the sense of Winnie Li and Patrick Sole.
Abstract: We generalize the Ihara-Selberg zeta function to hypergraphs in a natural way. Hashimoto's factorization results for biregular bipartite graphs apply, leading to exact factorizations. For $(d,r)$-regular hypergraphs, we show that a modified Riemann hypothesis is true if and only if the hypergraph is Ramanujan in the sense of Winnie Li and Patrick Sole. Finally, we give an example to show how the generalized zeta function can be applied to graphs to distinguish non-isomorphic graphs with the same Ihara-Selberg zeta function.
TL;DR: In this paper, it was shown that all metacyclic groups act with a distinguishing number at most 3, including all groups of square-free order, and that the distinguishing number of the action of the general linear group (GL_n(K) over a field K$ on the vector space K^n$ is 2 if K$ has at least n+1 elements.
Abstract: Let $G$ be a group acting faithfully on a set $X$. The distinguishing number of the action of $G$ on $X$, denoted $D_G(X)$, is the smallest number of colors such that there exists a coloring of $X$ where no nontrivial group element induces a color-preserving permutation of $X$. In this paper, we show that if $G$ is nilpotent of class $c$ or supersolvable of length $c$ then $G$ always acts with distinguishing number at most $c+1$. We obtain that all metacyclic groups act with distinguishing number at most 3; these include all groups of squarefree order. We also prove that the distinguishing number of the action of the general linear group $GL_n(K)$ over a field $K$ on the vector space $K^n$ is 2 if $K$ has at least $n+1$ elements.
TL;DR: A "hybrid method", dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux's method and singularity analysis theory.
Abstract: A "hybrid method", dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux's method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions—this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.
TL;DR: It is shown how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane and nine new infinite families of cubic partial cubes are found.
Abstract: We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.
TL;DR: Packing and covering problems in the finite metric space $(S_n,d_H)$ are surveyed, including a combination of both.
Abstract: Consider the symmetric group $S_n$ equipped with the Hamming metric $d_H$. Packing and covering problems in the finite metric space $(S_n,d_H)$ are surveyed, including a combination of both.
TL;DR: A simple combinatorial derivation of the moments of a single entry of a random unitary matrix is given, and it is deduced that the rescaled entries converge in moments to standard complex Gaussians.
Abstract: Let $U$ be a matrix chosen randomly, with respect to Haar measure, from the unitary group $U(d).$ For any $k \leq d,$ and any $k \times k$ submatrix $U_k$ of $U,$ we express the average value of $|{\rm Tr}(U_k)|^{2n}$ as a sum over partitions of $n$ with at most $k$ rows whose terms count certain standard and semistandard Young tableaux. We combine our formula with a variant of the Colour-Flavour Transformation of lattice gauge theory to give a combinatorial expansion of an interesting family of unitary matrix integrals. In addition, we give a simple combinatorial derivation of the moments of a single entry of a random unitary matrix, and hence deduce that the rescaled entries converge in moments to standard complex Gaussians. Our main tool is the Weingarten function for the unitary group.
TL;DR: An algorithm for finding a system of recurrence relations for the number of permutations of length $n$ that satisfy a certain set of conditions and an approach to describing such equations is proposed.
Abstract: We present an algorithm for finding a system of recurrence relations for the number of permutations of length $n$ that satisfy a certain set of conditions. A rewriting of these relations automatically gives a system of functional equations satisfied by the multivariate generating function that counts permutations by their length and the indices of the corresponding recurrence relations. We propose an approach to describing such equations. In several interesting cases the algorithm recovers and refines, in a unified way, results on $\tau$-avoiding permutations and permutations containing $\tau$ exactly once, where $\tau$ is any classical, generalized, and distanced pattern of length three.