Showing papers in "Electronic Journal of Combinatorics in 1999"
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TL;DR: Wilson et al. as mentioned in this paper showed that the weighted, directed spanning trees of any planar graph G$ can be put into a one-to-one weight-preserving correspondence with the perfect matchings of a related graph H$ on a directed weighted version of the cartesian lattice.
Abstract: In this article, Temperley's bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spanning trees (often called arborescences) of any planar graph $G$ can be put into a one-to-one weight-preserving correspondence with the perfect matchings of a related planar graph $H$. One special case of this result is a bijection between perfect matchings of the hexagonal honeycomb lattice and directed spanning trees of a triangular lattice. Another special case gives a correspondence between perfect matchings of the "square-octagon" lattice and directed weighted spanning trees on a directed weighted version of the cartesian lattice. In conjunction with results of Kenyon [1997b], our main theorem allows us to compute the measures of all cylinder events for random spanning trees on any (directed, weighted) planar graph. Conversely, in cases where the perfect matching model arises from a tree model, Wilson's algorithm allows us to quickly generate random samples of perfect matchings.
162 citations
TL;DR: It is proved that this can be achieved with $n=k^2", and the conjecture that asymptotically $n \sim (k/e)^2$ is the best achievable is proposed, and Noga Alon's conjecture that $n(\sigma)/2 $ is the threshold for random permutations is presented.
Abstract: Consider, for a permutation $\sigma \in {\cal S}_k$, the number $F(n,\sigma)$ of permutations in ${\cal S}_n$ which avoid $\sigma$ as a subpattern. The conjecture of Stanley and Wilf is that for every $\sigma$ there is a constant $c(\sigma) We also discuss $n$-permutations, containing all $\sigma \in {\cal S}_k$ as subpatterns. We prove that this can be achieved with $n=k^2$, we conjecture that asymptotically $n \sim (k/e)^2$ is the best achievable, and we present Noga Alon's conjecture that $n \sim (k/2)^2$ is the threshold for random permutations.
146 citations
TL;DR: It is proved that there exists a one-to-one correspondence between so-called fat linear codes over chain rings and multisets of points in projective Hjelmslev geometries, in the sense that semilinearly isomorphic codes correspond to equivalent multiset and vice versa.
Abstract: The aim of this paper is to develop a theory of linear codes over finite chain rings from a geometric viewpoint Generalizing a well-known result for linear codes over fields, we prove that there exists a one-to-one correspondence between so-called fat linear codes over chain rings and multisets of points in projective Hjelmslev geometries, in the sense that semilinearly isomorphic codes correspond to equivalent multisets and vice versa Using a selected class of multisets we show that certain MacDonald codes are linearly representable over nontrivial chain rings
98 citations
TL;DR: The values of Euler polynomials are related to certain "generalized Euler and class numbers" of D. Shanks, which have a combinatorial interpretation in terms of 3-signed permutations as defined by R. Ehrenborg and M. Readdy, and formulas for Springer and Shanks numbers are deduced in terms.
Abstract: Let $P_n$ and $Q_n$ be the polynomials obtained by repeated differentiation of the tangent and secant functions respectively. From the exponential generating functions of these polynomials we develop relations among their values, which are then applied to various numerical sequences which occur as values of the $P_n$ and $Q_n$. For example, $P_n(0)$ and $Q_n(0)$ are respectively the $n$th tangent and secant numbers, while $P_n(0)+Q_n(0)$ is the $n$th Andre number. The Andre numbers, along with the numbers $Q_n(1)$ and $P_n(1)-Q_n(1)$, are the Springer numbers of root systems of types $A_n$, $B_n$, and $D_n$ respectively, or alternatively (following V. I. Arnol'd) count the number of "snakes" of these types. We prove this for the latter two cases using combinatorial arguments. We relate the values of $P_n$ and $Q_n$ at $\sqrt3$ to certain "generalized Euler and class numbers" of D. Shanks, which have a combinatorial interpretation in terms of 3-signed permutations as defined by R. Ehrenborg and M. A. Readdy. Finally, we express the values of Euler polynomials at any rational argument in terms of $P_n$ and $Q_n$, and from this deduce formulas for Springer and Shanks numbers in terms of Euler polynomials.
69 citations
TL;DR: This work improves previous upper and lower bounds on the minimum size of an identifying code of an undirected graph when G is the two-dimensional square lattice and identifies the vertices of G if the neighbouring sets are nonempty and different.
Abstract: Let $G=(V,E)$ be an undirected graph. Let $C$ be a subset of vertices that we shall call a code. For any vertex $v\in V$, the neighbouring set $N(v,C)$ is the set of vertices of $C$ at distance at most one from $v$. We say that the code $C$ identifies the vertices of $G$ if the neighbouring sets $N(v,C), v\in V,$ are all nonempty and different. What is the smallest size of an identifying code $C$ ? We focus on the case when $G$ is the two-dimensional square lattice and improve previous upper and lower bounds on the minimum size of such a code.
61 citations
TL;DR: The continued fraction is found, in the form of a continued fraction, the generating function for the number of $(132)$-avoiding permutations that have a givenNumber of $(123)$ patterns, and it is shown how to extend this to permutation that have exactly one $( 132)$ pattern.
Abstract: We find, in the form of a continued fraction, the generating function for the number of $(132)$-avoiding permutations that have a given number of $(123)$ patterns, and show how to extend this to permutations that have exactly one $(132)$ pattern. We also find some properties of the continued fraction, which is similar to, though more general than, those that were studied by Ramanujan.
61 citations
TL;DR: The Grotzsch Theorem is extended to list colorings by proving that the clique hypergraph of every planar graph is 3-colorable and 4-choosability of ${\cal H}(G)$ is established for the class of locally planar graphs on arbitrary surfaces.
Abstract: In this paper, we extend the Grotzsch Theorem by proving that the clique hypergraph ${\cal H}(G)$ of every planar graph is 3-colorable. We also extend this result to list colorings by proving that ${\cal H}(G)$ is 4-choosable for every planar or projective planar graph $G$. Finally, 4-choosability of ${\cal H}(G)$ is established for the class of locally planar graphs on arbitrary surfaces.
59 citations
TL;DR: A description of the $c(\lambda,\mu,
u)$'s in terms of signed special rim hook tabloids and Littlewood-Richardson multitableaux is used and a lower bound is obtained on the stability of the multiplicity of the Kronecker product.
Abstract: F. Murnaghan observed a long time ago that the computation of the decompositon of the Kronecker product $\chi^{(n-a, \lambda_2, \dots )}\otimes \chi^{(n-b, \mu_2, \dots)}$ of two irreducible characters of the symmetric group into irreducibles depends only on $\overline\lambda=(\lambda_2,\dots )$ and $\overline\mu =(\mu_2,\dots )$, but not on $n$. In this note we prove a similar result: given three partitions $\lambda$, $\mu$, $
u$ of $n$ we obtain a lower bound on $n$, depending on $\overline\lambda$, $\overline\mu$, $\overline
u$, for the stability of the multiplicity $c(\lambda,\mu,
u)$ of $\chi^
u$ in $\chi^\lambda \otimes \chi^\mu$. Our proof is purely combinatorial. It uses a description of the $c(\lambda,\mu,
u)$'s in terms of signed special rim hook tabloids and Littlewood-Richardson multitableaux.
55 citations
TL;DR: A simple constructive version of Szemeredi's Regularity Lemma, based on the computation of singular values of matrices, is given.
Abstract: We give a simple constructive version of Szemeredi's Regularity Lemma, based on the computation of singular values of matrices.
53 citations
TL;DR: The quasi-triangular nature of these matrices gives a recursion for the Jack polynomials allowing for efficient computation and a determinantal formula for the Schur functions and a recursions for the Kostka numbers.
Abstract: We describe matrices whose determinants are the Jack polynomials expanded in terms of the monomial basis. The top row of such a matrix is a list of monomial functions, the entries of the sub-diagonal are of the form $-(r\alpha+s)$, with $r$ and $s \in {\bf N^+}$, the entries above the sub-diagonal are non-negative integers, and below all entries are 0. The quasi-triangular nature of these matrices gives a recursion for the Jack polynomials allowing for efficient computation. A specialization of these results yields a determinantal formula for the Schur functions and a recursion for the Kostka numbers.
53 citations
TL;DR: It is shown that if P(x) is a polynomial with nondecreasing, nonnegative coefficients, then the coefficient sequence of $P(x+1)$ is unimodal.
Abstract: We show that if $P(x)$ is a polynomial with nondecreasing, nonnegative coefficients, then the coefficient sequence of $P(x+1)$ is unimodal. Applications are given.
TL;DR: All the perfect 1-factorisations of K_{n,n} for $n\leq 9$ and count the Latin squares of order $9$ without proper subsquares without proper subrectangles are identified.
Abstract: A Latin square is pan-Hamiltonian if every pair of rows forms a single cycle. Such squares are related to perfect 1-factorisations of the complete bipartite graph. A square is atomic if every conjugate is pan-Hamiltonian. These squares are indivisible in a strong sense – they have no proper subrectangles. We give some existence results and a catalogue for small orders. In the process we identify all the perfect 1-factorisations of $K_{n,n}$ for $n\leq 9$, and count the Latin squares of order $9$ without proper subsquares.
TL;DR: In this paper, the authors give necessary conditions for the existence of a bicoloring with three color classes and give a multiplication theorem for Steiner triple systems with 3 color classes.
Abstract: A Steiner triple system has a bicoloring with $m$ color classes if the points are partitioned into $m$ subsets and the three points in every block are contained in exactly two of the color classes. In this paper we give necessary conditions for the existence of a bicoloring with 3 color classes and give a multiplication theorem for Steiner triple systems with 3 color classes. We also examine bicolorings with more than 3 color classes.
TL;DR: It is proved that for every integer $k\ge3$, for every (simple) series-parallel graph G, there exists a proper edge-coloring of G such that for Every edge of G, the color of e belongs to $L(e)$.
Abstract: It is proved that for every integer $k\ge3$, for every (simple) series-parallel graph $G$ with maximum degree at most $k$, and for every collection $(L(e):e\in E(G))$ of sets, each of size at least $k$, there exists a proper edge-coloring of $G$ such that for every edge $e\in E(G)$, the color of $e$ belongs to $L(e)$.
TL;DR: The main idea is to reduce the 'colored' problem to the case of usual random permutations using certain combinatorial results and elementary probabilistic arguments.
Abstract: We compute the limit distribution for the (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by Baik, Deift, and Johansson (math.CO/9810105). In the two–colored case our method provides a different proof of a similar result by Tracy and Widom about the longest increasing subsequences of signed permutations (math.CO/9811154). Our main idea is to reduce the 'colored' problem to the case of usual random permutations using certain combinatorial results and elementary probabilistic arguments.
TL;DR: It is shown that there is a weighing matrix W(4(q^m+q^{m-1}+\cdots+q+1)n^2,4q^mn^2) which includes two symmetric designs with the Ionin–type parameters which provides a new class of asymmetric designs.
Abstract: Let $4n^2$ be the order of a Bush-type Hadamard matrix with $q=(2n-1)^2$ a prime power. It is shown that there is a weighing matrix $$ W(4(q^m+q^{m-1}+\cdots+q+1)n^2,4q^mn^2) $$ which includes two symmetric designs with the Ionin–type parameters $$
u=4(q^m+q^{m-1}+\cdots+q+1)n^2,\;\;\; \kappa=q^m(2n^2-n), \;\;\; \lambda=q^m(n^2-n) $$ for every positive integer $m$. Noting that Bush–type Hadamard matrices of order $16n^2$ exist for all $n$ for which an Hadamard matrix of order $4n$ exist, this provides a new class of symmetric designs.
TL;DR: It is proved that for c 2.522 a random graph with n vertices and m=cn edges is not 3-colorable with probability 1-o(1) with bounds for non-k-colorability.
Abstract: We prove that for $c \geq 2.522$ a random graph with $n$ vertices and $m=cn$ edges is not 3-colorable with probability $1-o(1)$. Similar bounds for non-$k$-colorability are given for $k>3$.
TL;DR: The structure of SOMAs is studied, concentrating on how SOMAs can decompose, and the use of computational group theory and graph theory in the discovery and classification of SOMAs is reported on.
Abstract: Let $k\ge0$ and $n\ge2$ be integers. A SOMA, or more specifically a SOMA$(k,n)$, is an $n\times n$ array $A$, whose entries are $k$-subsets of a $kn$-set $\Omega$, such that each element of $\Omega$ occurs exactly once in each row and exactly once in each column of $A$, and no 2-subset of $\Omega$ is contained in more than one entry of $A$. A SOMA$(k,n)$ can be constructed by superposing $k$ mutually orthogonal Latin squares of order $n$ with pairwise disjoint symbol-sets, and so a SOMA$(k,n)$ can be seen as a generalization of $k$ mutually orthogonal Latin squares of order $n$. In this paper we first study the structure of SOMAs, concentrating on how SOMAs can decompose. We then report on the use of computational group theory and graph theory in the discovery and classification of SOMAs. In particular, we discover and classify SOMA$(3,10)$s with certain properties, and discover two SOMA$(4,14)$s (SOMAs with these parameters were previously unknown to exist). Some of the newly discovered SOMA$(3,10)$s come from superposing a Latin square of order 10 on a SOMA$(2,10)$.
TL;DR: It is proven for all $n\ge 2, $d\ge1, and $k>d$ that $C^{n}_{d,k}=0$ if and only if $k=d+1$.
Abstract: For integers $d$ and $k$ satisfying $0 \le d \le k$, a binary sequence is said to satisfy a one-dimensional $(d,k)$ run length constraint if there are never more than $k$ zeros in a row, and if between any two ones there are at least $d$ zeros. For $n\geq 1$, the $n$-dimensional $(d,k)$-constrained capacity is defined as $$C^{n}_{d,k} = \lim_{m_1,m_2,\ldots,m_n\rightarrow\infty} {{\log_2 N_{m_1,m_2,\ldots ,m_n}^{(n; d,k)}} \over {m_1 m_2\cdots m_n}} $$ where $N_{m_1,m_2,\ldots ,m_n}^{(n; d,k)}$ denotes the number of $m_1\times m_2\times\cdots\times m_n$ $n$-dimensional binary rectangular patterns that satisfy the one-dimensional $(d,k)$ run length constraint in the direction of every coordinate axis. It is proven for all $n\ge 2$, $d\ge1$, and $k>d$ that $C^{n}_{d,k}=0$ if and only if $k=d+1$. Also, it is proven for every $d\geq 0$ and $k\geq d$ that $\lim_{n\rightarrow\infty}C^{n}_{d,k}=0$ if and only if $k\le 2d$.
TL;DR: The case of the common vertex being the center of the hexagon solves a problem posed by Propp.
Abstract: We compute the number of rhombus tilings of a hexagon with sides $n$, $n$, $N$, $n$, $n$, $N$, where two triangles on the symmetry axis touching in one vertex are removed. The case of the common vertex being the center of the hexagon solves a problem posed by Propp.
TL;DR: It is shown how to count tilings of Aztec diamonds and hexagons with defects using determinants, and solutions to open problems 1, 2, and 10 in James Propp's list of problems on enumeration of matchings are obtained.
Abstract: We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In several cases these determinants can be evaluated in closed form. In particular, we obtain solutions to open problems 1, 2, and 10 in James Propp's list of problems on enumeration of matchings.
TL;DR: This theorem and another local limit theorem which is useful for conditioning are applied to various combinatorial enumeration problems that involve multivariate Lagrange inversion.
Abstract: An asymptotic estimate is given for the coefficients of products of large powers of generating functions. This theorem and another local limit theorem which is useful for conditioning are applied to various combinatorial enumeration problems that involve multivariate Lagrange inversion.
TL;DR: It is shown that the function f is uniformly continuous under the $2-adic metric, and thus extends to a function on all of $Z$ and satisfies the functional equation f(-1-n) = \pm f(n)$, where the sign is positive iff $n \equiv 0,3 \pmod{4}$.
Abstract: We study the $2$-adic behavior of the number of domino tilings of a $2n \times 2n$ square as $n$ varies. It was previously known that this number was of the form $2^nf(n)^2$, where $f(n)$ is an odd, positive integer. We show that the function $f$ is uniformly continuous under the $2$-adic metric, and thus extends to a function on all of $Z$. The extension satisfies the functional equation $f(-1-n) = \pm f(n)$, where the sign is positive iff $n \equiv 0,3 \pmod{4}$.
TL;DR: Graham's construction is improved and generalize Wilf's note, and it is shown that the 12-digit pair $$(a,b)= (407389224418,76343678551)$$ also defines such a sequence.
Abstract: In 1964, Ronald Graham proved that there exist relatively prime natural numbers $a$ and $b$ such that the sequence $\{A_n\}$ defined by $$ {A}_{n} =A_{n-1}+A_{n-2}\qquad (n\ge 2;A_0=a,A_1=b)$$ contains no prime numbers, and constructed a 34-digit pair satisfying this condition. In 1990, Donald Knuth found a 17-digit pair satisfying the same conditions. That same year, noting an improvement to Knuth's computation, Herbert Wilf found a yet smaller 17-digit pair. Here we improve Graham's construction and generalize Wilf's note, and show that the 12-digit pair $$(a,b)= (407389224418,76343678551)$$ also defines such a sequence.
TL;DR: It is shown that all posets with local actions induced by labellings have symmetric chain decompositions and provide $R^* S-labellings for the type B and D noncrossing partition lattices, answering a question of Stanley.
Abstract: A symmetric group action on the maximal chains in a finite, ranked poset is local if the adjacent transpositions act in such a way that $(i,i+1)$ sends each maximal chain either to itself or to one differing only at rank $i$. We prove that when $S_n$ acts locally on a lattice, each orbit considered as a subposet is a product of chains. We also show that all posets with local actions induced by labellings known as $R^* S$-labellings have symmetric chain decompositions and provide $R^* S$-labellings for the type B and D noncrossing partition lattices, answering a question of Stanley.
TL;DR: In this paper, it was shown that for any connected graph that is not a path, the inequality of the maximum degree δ(k+1 + 2 Δ(k)-k-2 holds.
Abstract: Let $\Delta_k$ denote the maximum degree of the $k^{\rm th}$ iterated line graph $L^k(G)$. For any connected graph $G$ that is not a path, the inequality $\Delta_{k+1}\leq 2\Delta_k-2$ holds. Niepel, Knor, and Soltes have conjectured that there exists an integer $K$ such that, for all $k\geq K$, equality holds; that is, the maximum degree $\Delta_k$ attains the greatest possible growth. We prove this conjecture using induced subgraphs of maximum degree vertices and locally maximum vertices.
TL;DR: A combinatorial interpretation of the total area of elevated Schroder paths is given, by first introducing sets of unrestricted paths whose cardinality also satisfies the recurrence relation and then establishing a bijection between the set of these paths and theset of triangles constituting thetotal area.
Abstract: An elevated Schroder path is a lattice path that uses the steps $(1,1)$, $(1,-1)$, and $(2,0)$, that begins and ends on the $x$-axis, and that remains strictly above the $x$-axis otherwise The total area of elevated Schroder paths of length $2n+2$ satisfies the recurrence $f_{n+1}=6f_n-f_{n-1}$, $n \geq 2$, with the initial conditions $f_0=1$, $f_1=7$ A combinatorial interpretation of this recurrence is given, by first introducing sets of unrestricted paths whose cardinality also satisfies the recurrence relation and then establishing a bijection between the set of these paths and the set of triangles constituting the total area of elevated Schroder paths
TL;DR: It is proved that the threshold for a random graph to have a $k$-core is equal to thereshold for having a subgraph which meets a necessary condition of Gallai for being $k-critical.
Abstract: We prove that the threshold for a random graph to have a $k$-core is equal to the threshold for having a subgraph which meets a necessary condition of Gallai for being $k$-critical.
TL;DR: It is proved that every locally finite subset of $\mathbb{R}$ is reconstructible from the multiset of its 3-subsets, given up to translation.
Abstract: We consider the problem of reconstructing a set of real numbers up to translation from the multiset of its subsets of fixed size, given up to translation. This is impossible in general: for instance almost all subsets of $\mathbb{Z}$ contain infinitely many translates of every finite subset of $\mathbb{Z}$. We therefore restrict our attention to subsets of $\mathbb{R}$ which are locally finite ; those which contain only finitely many translates of any given finite set of size at least 2. We prove that every locally finite subset of $\mathbb{R}$ is reconstructible from the multiset of its 3-subsets, given up to translation.
TL;DR: An elementary proof that the lattice point counts in the interior and closure of such a vector- dilated simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law is given.
Abstract: We generalize Ehrhart's idea ((Eh)) of counting lattice points in dilated rational poly- topes: Given a rational simplex, that is, an n-dimensional polytope with n + 1 rational vertices, we use its description as the intersection of n + 1 halfspaces, which determine the facets of the simplex. Instead of just a single dilation factor, we allow dierent dilation factors for each of these facets. We give an elementary proof that the lattice point counts in the interior and closure of such a vector- dilated simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classical reciprocity law for rational polytopes ((Ma), (Mc), (St)). As an example, we derive a lattice point count formula for a rectangular rational triangle, which enables us to compute the number of lattice points inside any rational polygon.