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By using totally isotropic subspaces in an orthogonal space Omega^{+}(2i,2), several infinite families of packings of 2^k-dimensional subspaces of real 2^i-dimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and links this problem with Barnes-Wall lattices, Kerdock sets and quantum-error-correcting codes. | A Group-Theoretic Framework for the Construction of Packings in
Grassmannian Spaces | 13,700 |
This paper addresses the question: how should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N, n, m are described, as well as a reformulation of the problem that was suggested by these computations. The reformulation gives a way to describe n-dimensional subspaces of m-space as points on a sphere in dimension (m-1)(m+2)/2, which provides a (usually) lower-dimensional representation than the Pluecker embedding, and leads to a proof that many of the new packings are optimal. The results have applications to the graphical display of multi-dimensional data via Asimov's "Grand Tour" method. | Packing Lines, Planes, etc.: Packings in Grassmannian Space | 13,701 |
The diagram of a 132-avoiding permutation can easily be characterized: it is simply the diagram of a partition. Based on this fact, we present a new bijection between 132-avoiding and 321-avoiding permutations. We will show that this bijection translates the correspondences between these permutations and Dyck paths given by Krattenthaler and by Billey-Jockusch-Stanley, respectively, to each other. Moreover, the diagram approach yields simple proofs for some enumerative results concerning forbidden patterns in 132-avoiding permutations. | On the diagram of 132-avoiding permutations | 13,702 |
Several authors have examined connections between permutations which avoid 132, continued fractions, and Chebyshev polynomials of the second kind. In this paper we prove analogues of some of these results for permutations which avoid 1243 and 2143. Using tools developed to prove these analogues, we give enumerations and generating functions for permutations which avoid 1243, 2143, and certain additional patterns. We also give generating functions for permutations which avoid 1243 and 2143 and contain certain additional patterns exactly once. In all cases we express these generating functions in terms of Chebyshev polynomials of the second kind. | Permutations Which Avoid 1243 and 2143, Continued Fractions, and
Chebyshev Polynomials | 13,703 |
I give a survey of different combinatorial forms of alternating-sign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as corner-sum matrices, height-function matrices, three-colorings, monotone triangles, tetrahedral order ideals, square ice, gasket-and-basket tilings and full packings of loops. | The many faces of alternating-sign matrices | 13,704 |
From any given sequence of finite or infinite graphs, a nonstandard graph is constructed. The procedure is similar to an ultrapower construction of an internal set from a sequence of subsets of the real line, but now the individual entities are the vertices of the graphs instead of real numbers. The transfer principle is then invoked to extend several graph-theoretic results to the nonstandard case. After incidences and adjacencies between nonstandard vertices are defined, several formulas regarding numbers of vertices and edges, and nonstandard versions of Eulerian graphs, Hamiltonian graphs, and a coloring theorem are established for these nonstandard graphs. | Nonstandard Graphs | 13,705 |
Consider the algebra Q<<x_1,x_2,...>> of formal power series in countably many noncommuting variables over the rationals. The subalgebra Pi(x_1,x_2,...) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as will as investigating their properties. | Symmetric Functions in Noncommuting Variables | 13,706 |
In 1986, Oliver Pretzel studied the set of orientations of a connected finite graph $G$ and showed that any two such orientations having the same flow-difference around all closed loops can be obtained from one another by a succession of local moves of a simple type. Here I show that the set of orientations of $G$ having the same flow-differences around all closed loops can be given the structure of a distributive lattice. When the graph is drawn on the plane, a dual version of the construction puts a distributive lattice structure on the set of orientations of $G$ having the same indegrees at all vertices. In both settings, adjacent lattice-elements are related by simple local moves. This construction unifies earlier, similar constructions in combinatorics and statistical mechanics. It also gives rise to a lattice structure on spanning trees that seems to be new. | Lattice structure for orientations of graphs | 13,707 |
We study the number of meromorphic functions on a Riemann surface with given critical values and prescribed multiplicities of critical points and values. When the Riemann surface is $\CP^1$ and the function is a polynomial, we give an elementary way of finding this number. In the general case, we show that, as the multiplicities of critical points tend to infinity, the asymptotic for the number of meromorphic functions is given by the volume of some space of graphs glued from circles. We express this volume as a matrix integral. | Counting Meromorphic Functions with Critical Points of Large
Multiplicities | 13,708 |
Machines whose main purpose is to permute and sort data are studied. The sets of permutations that can arise are analysed by means of finite automata and avoided pattern techniques. Conditions are given for these sets being enumerated by rational generating functions. | Regular closed classes of permutations | 13,709 |
We study a) the limit of the ratio of two consecutive terms in such a sequence and b) the limit of the ratio of two terms in which one has a lag equal to 2. In the general case limit a) does not exist but we have two limiting values depending on the parity of the index. And these limits depend on the initial conditions. Limit b) exists and does not depend on the the initial conditions. Finally we seek the set of initial conditions for which this limit exists (that is, the two limiting values coincide) and obtain this limit. | Degenerated third order linear recurrences | 13,710 |
The theme of this article is the algebraic combinatorics of leaf-labeled rooted binary trees and forests of such trees. The structure of a Hopf operad is defined on the vector spaces spanned by forests of leaf-labeled, rooted, binary trees. An explicit formula for the coproduct and its dual product is given, using a poset on forests. | A Hopf operad of forests of binary trees and related finite-dimensional
algebras | 13,711 |
We introduce a new graph polynomial in two variables. This ``interlace'' polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the significant differences are that our expansion is over vertex rather than edge subsets, and the rank and nullity employed are those of an adjacency matrix rather than an incidence matrix. The second computation is by a three-term reduction formula involving a graph pivot; the pivot arose previously in the study of interlacement and Euler circuits in four-regular graphs. We consider a few properties and specializations of the two-variable interlace polynomial. One specialization, the ``vertex-nullity interlace polynomial'', is the single-variable interlace graph polynomial we studied previously, closely related to the Tutte-Martin polynomial on isotropic systems previously considered by Bouchet. Another, the ``vertex-rank interlace polynomial'', is equally interesting. Yet another specialization of the two-variable polynomial is the independent-set polynomial. | A Two-Variable Interlace Polynomial | 13,712 |
Represented Coxeter matroids of types $C_n$ and $D_n$, that is, symplectic and orthogonal matroids arising from totally isotropic subspaces of symplectic or (even-dimensional) orthogonal spaces, may also be represented in buildings of type $C_n$ and $D_n$, respectively. Indeed, the particular buildings involved are those arising from the flags or oriflammes, respectively, of totally isotropic subspaces. There are also buildings of type $B_n$ arising from flags of totally isotropic subspaces in odd-dimensional orthogonal space. Coxeter matroids of type $B_n$ are the same as those of type $C_n$ (since they depend only upon the reflection group, not the root system). However, buildings of type $B_n$ are distinct from those of the other types. The matroids representable in odd dimensional orthogonal space (and therefore in the building of type $B_n$) turn out to be a special case of symplectic (flag) matroids, those whose top component, or Lagrangian matroid, is a union of two Lagrangian orthogonal matroids. These two matroids are called a Lagrangian pair, and they are the combinatorial manifestation of the ``fork'' at the top of an oriflamme (or of the fork at the end of the Coxeter diagram of $D_n$). Here we give a number of equivalent characterizations of Lagrangian pairs, and prove some rather strong properties of them. | Lagrangian Pairs and Lagrangian Orthogonal Matroids | 13,713 |
An ordinal-valued metric taking its values in the set of all countable ordinals can be assigned to a metrizable set of nodes in a transfinite graph. Then, a variety of results concerning nodal eccentricities, radii, diameters, centers, peripheries, and blocks can be extended to transfinite graphs. | Ordinal Distances in Transfinite Graphs | 13,714 |
Fix a rectangular Young diagram R, and consider all the products of Schur functions s(mu) s(mu^c), where mu and mu^c run over all (unordered) pairs of partitions which are complementary with respect to R. Theorem: The self-complementary products, s(mu)^2 where mu=mu^c, are linearly independent of all other s(mu) s(mu^c). Conjecture: The products s(mu) s(mu^c) are all linearly independent. | Linearly Independent Products of Rectangularly Complementary Schur
Functions | 13,715 |
It is shown explicitly how self-similar graphs can be obtained as `blow-up' constructions of finite cell graphs $\hat C$. This yields a larger family of graphs than the graphs obtained by discretising continuous self-similar fractals. For a class of symmetrically self-similar graphs we study the simple random walk on a cell graph $\hat C$, starting in a vertex $v$ of the boundary of $\hat C$. It is proved that the expected number of returns to $v$ before hitting another vertex in the boundary coincides with the resistance scaling factor. Using techniques from complex rational iteration and singularity analysis for Green functions we compute the asymptotic behaviour of the $n$-step transition probabilities of the simple random walk on the whole graph. The results of Grabner and Woess for the Sierpi\'nski graph are generalised to the class of symmetrically self-similar graphs and at the same time the error term of the asymptotic expression is improved. Finally we present a criterion for the occurrence of oscillating phenomena of the $n$-step transition probabilities. | Asymptotics of the transition probabilities of the simple random walk on
self-similar graphs | 13,716 |
We establish some identities relating two sequences that are, as explained, related to the Tribonacci sequence. One of these sequences bears the same resemblance to the Tribonacci sequence as the Lucas sequence does to the Fibonacci sequence. Defining a matrix that we call Tribomatrix, which extends the Fibonacci matrix, we see that the other sequence is related to the sum of the determinants of the 2nd order principal minors of this matrix. | Identities for Tribonacci-related sequences | 13,717 |
Egge and Mansour have recently studied permutations which avoid 1243 and 2143 regarding the occurrence of certain additional patterns. Some of the open questions related to their work can easily be answered by using permutation diagrams. Like for 132-avoiding permutations the diagram approach gives insights into the structure of {1243,2143}-avoiding permutations that yield simple proofs for some enumerative results concerning forbidden patterns in such permutations. | On the diagram of Schroeder permutations | 13,718 |
Simion conjectured the unimodality of a sequence counting lattice paths in a grid with a Ferrers diagram removed from the northwest corner. Recently, Hildebrand and then Wang proved the stronger result that this sequence is actually log concave. Both proofs were mainly algebraic in nature. We give two combinatorial proofs of this theorem. | Two injective proofs of a conjecture of Simion | 13,719 |
In this thesis, we apply the stack sorting operator to $r$-permutations and construct the functional equation for the generating function of two-stack-sortable $k$-tuple $r$-permutations counted by descents by using a factorization similar to Zeilberger's. We solve the functional equation and give explicit formulas for the number of two-stack-sortable $r$-permutations. | Generalizations of two-stack-sortable permutations | 13,720 |
In [Kit1] Kitaev discussed simultaneous avoidance of two 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. In three essentially different cases, the numbers of such $n$-permutations are $2^{n-1}$, the number of involutions in $\mathcal{S}_n$, and $2E_n$, where $E_n$ is the $n$-th Euler number. In this paper we give recurrence relations for the remaining three essentially different cases. To complete the descriptions in [Kit3] and [KitMans], we consider avoidance of a pattern of the form $x-y-z$ (a classical 3-pattern) and beginning or ending with an increasing or decreasing pattern. Moreover, we generalize this problem: we demand that a permutation must avoid a 3-pattern, begin with a certain pattern and end with a certain pattern simultaneously. We find the number of such permutations in case of avoiding an arbitrary generalized 3-pattern and beginning and ending with increasing or decreasing patterns. | On multi-avoidance of generalized patterns | 13,721 |
We show how the set of Dyck paths of length 2n naturally gives rise to a matroid, which we call the "Catalan matroid" C_n. We describe this matroid in detail; among several other results, we show that C_n is self-dual, it is representable over the rationals but not over finite fields F_q with q < n-1, and it has a nice Tutte polynomial. We then generalize our construction to obtain a family of matroids, which we call "shifted matroids". They arose independently and almost simultaneously in the work of Klivans, who showed that they are precisely the matroids whose independence complex is a shifted complex. | The Catalan matroid | 13,722 |
We present matrix identities which yield respectively the Jordan canonical form of the Pascal matrix P_n = (i -1 choose j -1)_{1 <= i,j <= n} modulo a prime, the eigenvectors of (i choose j)_{1 <= i,j <= n}, and the Smith normal form of powers of P_n - I_n. | Jordan and Smith forms of Pascal-related matrices | 13,723 |
A permutation $\pi$ is said to be {\em Dumont permutations of the first kind} if each even integer in $\pi$ must be followed by a smaller integer, and each odd integer is either followed by a larger integer or is the last element of $\pi$ (see, for example, \cite{Z}). In \cite{D} Dumont showed that certain classes of permutations on $n$ letters are counted by the Genocchi numbers. In particular, Dumont showed that the $(n+1)$st Genocchi number is the number of Dummont permutations of the first kind on $2n$ letters. In this paper we study the number of Dumont permutations of the first kind on $n$ letters avoiding the pattern 132 and avoiding (or containing exactly once) an arbitrary pattern on $k$ letters. In several interesting cases the generating function depends only on $k$. | Restricted 132-Dumont permutations | 13,724 |
Circular permutations on {1,2,...,n} that avoid a given pattern correspond to ordinary (linear) permutations that end with n and avoid all cyclic rotations of the pattern. Three letter patterns are all but unavoidable in circular permutations and here we give explicit formulas for the number of circular permutations that avoid one four letter pattern. In the three essentially distinct cases, the counts are as follows: the Fibonacci number F_{2n-3} for the pattern 1324, 2^{n-1}-(n-1) for 1342, and 2^{n}+1-2n-{n}choose{3} for 1234. | Pattern avoidance in circular permutations | 13,725 |
Recently, Kitaev [Ki2] introduced partially ordered generalized patterns (POGPs) in the symmetric group, which further generalize the generalized permutation patterns introduced by Babson and Steingr\'imsson [BS]. A POGP p is a GP some of whose letters are incomparable. In this paper, we study the generating functions (g.f.) for the number of k-ary words avoiding some POGPs. We give analogues, extend and generalize several known results, as well as get some new results. In particular, we give the g.f. for the entire distribution of the maximum number of non-overlapping occurrences of a pattern p with no hyphens (that allowed to have repetition of letters), provided we know the g.f. for the number of k-ary words that avoid p. | Partially Ordered generalized patterns and k-ary words | 13,726 |
A permutation is said to be \emph{alternating} if it starts with rise and then descents and rises come in turn. In this paper we study the generating function for the number of alternating permutations on $n$ letters that avoid or contain exactly once 132 and also avoid or contain exactly once an arbitrary pattern on $k$ letters. In several interesting cases the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind. | Restricted 132-alternating permutations and Chebyshev polynomials | 13,727 |
This version is similar to math.CO/0210113. We've changed Conjectures 1.1 and 1.2 so that they cover arbitrary graphs(digraphs). Let G be an arbitrary graph(digraph). Then - in polynomial time - either an algorithm obtains a hamilton circuit(cycle)or else the algorithm points to at least one vertex that cannot belong to any hamilton circuit(cycle) of G. We give criteria for determining which vertices should be examined. | H-admissible permutations and the HCP | 13,728 |
We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1-...-1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In particular, we find the number of occurrences of the patterns 1-2, 2-1, 12, 21, 123 and 1-1-...-1 in the words obtained by iterations of the morphism 1->123, 2->13, 3->2, which is a classical example of a morphism generating a nonrepetitive sequence. | Counting the occurrences of generalized patterns in words generated by a
morphism | 13,729 |
A class of associative (super) algebras is presented, which naturally generalize both the symmetric algebra $Sym(V)$ and the wedge algebra $\wedge (V)$, where $V$ is a vector-space. These algebras are in a bijection with those subsets of the set of the partitions which are closed under inclusions of partitions. We study the rate of growth of these algebras, then characterize the case where these algebras satisfy polynomial identities. | On a class of algebras defined by partitions | 13,730 |
We consider $m$-th order linear recurrences that can be thought of as generalizations of the Lucas sequence. We exploit some interplay with matrices that again can be considered generalizations of the Fibonacci matrix. We introduce the definition of reflected sequence and inverted sequence and we establish some relationship between the coefficients of the Cayley-Hamilton equation for these matrices and the introduced sequences. | Polymatrix and generalized polynacci numbers | 13,731 |
Let n,p,k be three positive integers. We prove that the numbers binomial (n,k) 3F2 (1-k, -p, p-n ; 1, 1-n ; 1) are positive integers which generalize the classical binomial coefficients. We give two generating functions for these integers, and a straightforward application. | A new family of positive integers | 13,732 |
Explicit algorithms are developed for constructing odd order n pandiagonal latin cubes in 3 and 4 dimensions, and these are used to construct pandiagonal magic cubes and 4 dimensional hypercubes, respectively. It is established that these structures exist in 3 dimensions only for odd orders n not less than 11, and in 4 dimensions only for odd orders n not less than 17. Estimates for the number of such structures as functions of n in 3 and 4 dimensions are also presented. | Odd Order Pandiagonal Latin and Magic Cubes in Three and Four Dimensions | 13,733 |
Let $TT_k$ denote the transitive tournament on $k$ vertices. Let $TT(h,k)$ denote the graph obtained from $TT_k$ by replacing each vertex with an independent set of size $h \geq 1$. The following result is proved: Let $c_2=1/2$, $c_3=5/6$ and $c_k=1-2^{-k-\log k}$ for $k \geq 4$. For every $\epsilon > 0$ there exists $N=N(\epsilon,h,k)$ such that for every undirected graph $G$ with $n > N$ vertices and with $\delta(G) \geq c_kn$, every orientation of $G$ contains vertex disjoint copies of $TT(h,k)$ that cover all but at most $\epsilon n$ vertices. In the cases $k=2$ and $k=3$ the result is asymptotically tight. For $k \geq 4$, $c_k$ cannot be improved to less than $1-2^{-0.5k(1+o(1))}$. | Tiling transitive tournaments and their blow-ups | 13,734 |
Let $F=\{H_1,...,H_k\}$ be a family of graphs. A graph $G$ with $m$ edges is called {\em totally $F$-decomposable} if for {\em every} linear combination of the form $\alpha_1 e(H_1) + ... + \alpha_k e(H_k) = m$ where each $\alpha_i$ is a nonnegative integer, there is a coloring of the edges of $G$ with $\alpha_1+...+\alpha_k$ colors such that exactly $\alpha_i$ color classes induce each a copy of $H_i$, for $i=1,...,k$. We prove that if $F$ is any fixed family of trees then $\log n/n$ is a sharp threshold function for the property that the random graph $G(n,p)$ is totally $F$-decomposable. In particular, if $H$ is a tree, then $\log n/n$ is a sharp threshold function for the property that $G(n,p)$ contains $\lfloor e(G)/e(H) \rfloor$ edge-disjoint copies of $H$. | Families of trees decompose the random graph in any arbitrary way | 13,735 |
The number of stable sets of cardinality $k$ in graph $G$ is the $k$-th coefficient of the independence polynomial of $G$ (I. Gutman and F. Harary, 1983). In 1990, Y. O. Hamidoune proved that for any claw-free graph, its independence polynomial is unimodal, i.e., there exists a coefficient $k$ such that the part of the sequence of coefficients from the first to $k$-th is non-decreasing while the second part of coefficients is non-increasing. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erd\"{o}s (1987) asked whether for trees (or perhaps forests) the independence polynomial is unimodal. J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured that it is true for any well-covered graph (a graph whose all maximal independent sets have the same size). V. E. Levit and E. Mandrescu (1999) demonstrated that every well-covered tree can be obtained as a join of a number of well-covered spiders, where a spider is a tree having at most one vertex of degree at least three. In this paper we show that the independence polynomial of any well-covered spider is unimodal. In addition, we introduce some graph transformations respecting independence polynomials. They allow us to reduce several types of well-covered trees to claw-free graphs, and, consequently, to prove that their independence polynomials are unimodal. | On Unimodality of Independence Polynomials of some Well-Covered Trees | 13,736 |
Let $A$ be a commutative $k$-algebra over a field of $k$ and $\Xi$ a linear operator defined on $A$. We define a family of $A$-valued invariants $\Psi$ for finite rooted forests by a recurrent algorithm using the operator $\Xi$ and show that the invariant $\Psi$ distinguishes rooted forests if (and only if) it distinguishes rooted trees $T$, and if (and only if) it is {\it finer} than the quantity $\alpha (T)=|\text{Aut}(T)|$ of rooted trees $T$. We also consider the generating function $U(q)=\sum_{n=1}^\infty U_n q^n$ with $U_n =\sum_{T\in \bT_n} \frac 1{\alpha (T)} \Psi (T)$, where $\bT_n$ is the set of rooted trees with $n$ vertices. We show that the generating function $U(q)$ satisfies the equation $\Xi \exp U(q)= q^{-1} U(q)$. Consequently, we get a recurrent formula for $U_n$ $(n\geq 1)$, namely, $U_1=\Xi(1)$ and $U_n =\Xi S_{n-1}(U_1, U_2, >..., U_{n-1})$ for any $n\geq 2$, where $S_n(x_1, x_2, ...)$ $(n\in \bN)$ are the elementary Schur polynomials. We also show that the (strict) order polynomials and two well known quasi-symmetric function invariants of rooted forests are in the family of invariants $\Psi$ and derive some consequences about these well-known invariants from our general results on $\Psi$. Finally, we generalize the invariant $\Psi$ to labeled planar forests and discuss its certain relations with the Hopf algebra $\mathcal H_{P, R}^D$ in \cite{F} spanned by labeled planar forests. | A Family of Invariants of Rooted Forests | 13,737 |
Let P be a poset with elements 1,2,...,n. We say that P is sign-balanced if exactly half the linear extensions of P (regarded as permutations of 1,2,...,n) are even permutations, i.e., have an even number of inversions. This concept first arose in the work of Frank Ruskey, who was interested in the efficient generation of all linear extensions of P. We survey a number of techniques for showing that posets are sign-balanced, and more generally, computing their "imbalance." There are close connections with domino tilings and, for certain posets, a "domino generalization" of Schur functions due to Carre and Leclerc. We also say that P is maj-balanced if exactly half the linear extensions of P have even major index. We discuss some similarities and some differences between sign-balanced and maj-balanced posets. | Some remarks on sign-balanced and maj-balanced posets | 13,738 |
A survey of three recent developments in algebraic combinatorics: (1) the Laurent phenomenon, (2) Gromov-Witten invariants and toric Schur functions, and (3) toric h-vectors and intersection cohomology. This paper is a continuation of "Recent progress in algebraic combinatorics" (math.CO/0010218), which dealt with three other topics. | Recent developments in algebraic combinatorics | 13,739 |
It is known that a graded lattice of rank n is supersolvable if and only if it has an EL-labelling where the labels along any maximal chain are exactly the numbers 1,2,...,n without repetition. These labellings are called S_n EL-labellings, and having such a labelling is also equivalent to possessing a maximal chain of left modular elements. In the case of an ungraded lattice, there is a natural extension of S_n EL-labellings, called interpolating labellings. We show that admitting an interpolating labelling is again equivalent to possessing a maximal chain of left modular elements. Furthermore, we work in the setting of a general bounded poset as all the above results generalize to this case. We conclude by applying our results to show that the lattice of non-straddling partitions, which is not graded in general, has a maximal chain of left modular elements. | Poset Edge-Labellings and Left Modularity | 13,740 |
Let $C^{2k}_r$ be the $2k$-uniform hypergraph obtained by letting $P_1,...,P_r$ be pairwise disjoint sets of size $k$ and taking as edges all sets $P_i \cup P_j$ with $i \neq j$. This can be thought of as the `$k$-expansion' of the complete graph $K_r$: each vertex has been replaced with a set of size $k$. We determine the exact Turan number of $C^{2k}_3$ and the corresponding extremal hypergraph, thus confirming a conjecture of Frankl. Sidorenko has given an upper bound of $(r-2) / (r-1)$ for the Tur\'an density of $C^{2k}_r$ for any $r$, and a construction establishing a matching lower bound when $r$ is of the form $2^p + 1$. We show that when $r = 2^p + 1$, any $C^4_r$-free hypergraph of density $(r-2)/(r-1) - o(1)$ looks approximately like Sidorenko's construction. On the other hand, when $r$ is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Tur\'an density of $C^4_r$ to $(r-2)/(r-1) - c(r)$, where $c(r)$ is a constant depending only on $r$. The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal-Katona theorem and properties of Krawtchouck polynomials. | On a hypergraph Turan problem of Frankl | 13,741 |
We define a natural class of graphs by generalizing prior notions of visibility, allowing the representing regions and sightlines to be arbitrary. We consider mainly the case of compact connected representing regions, proving two results giving necessary properties of visibility graphs, and giving some examples of classes of graphs that can be so represented. Finally, we give some applications of the concept, and we provide potential avenues for future research in the area. | A general notion of visiblity graphs | 13,742 |
Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0,0) to (m,r) and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal matroid, which we call a lattice path matroid. We consider a variety of enumerative aspects of these matroids and we study three important matroid invariants, namely the Tutte polynomial and, for special types of lattice path matroids, the characteristic polynomial and the beta invariant. In particular, we show that the Tutte polynomial is the generating function for two basic lattice path statistics and we show that certain sequences of lattice path matroids give rise to sequences of Tutte polynomials for which there are relatively simple generating functions. We show that Tutte polynomials of lattice path matroids can be computed in polynomial time. Also, we obtain a new result about lattice paths from an analysis of the beta invariant of certain lattice path matroids. | Lattice path matroids: enumerative aspects and Tutte polynomials | 13,743 |
We study the generating function for the number of even (or odd) permutations on n letters containing exactly $r\gs0$ occurrences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in $S_{2r}$. | Counting occurrences of 132 in an even permutation | 13,744 |
A connection between permutations that avoid 4231 and a certain queueing discipline is established. It is proved that a more restrictive queueing discipline corresponds to avoiding both 4231 and 42513, and enumeration results for such permutations are given. | Restricted permutations and queue jumping | 13,745 |
We prove a conjecture due to Holroyd and Johnson that an analogue of the Erdos-Ko-Rado theorem holds for k-separated sets. In particular this determines the independence number of the vertex-critical subgraph of the Kneser graph identified by Schrijver, the collection of separated sets. | Intersecting Families of Separated Sets | 13,746 |
The Narayana numbers are $N(n,k) = {1 \over n}{n \choose k}{n \choose {k+1}}$. There are several natural statistics on Dyck paths with a distribution given by N(n,k). We show the equidistribution of Narayana statistics by computing the flag h-vector of $J({\bf 2} \times {\bf n})$ in different ways. In the process we discover new Narayana statistics and provide co-statistics for which the Narayana statistics in question have a distribution given by F\"urlinger and Hofbauers q-Narayana numbers. We also interpret the h-vector in terms of semi-standard Young tableaux, which enables us to express the q-Narayana numbers in terms of Schur functions. | q-Narayana numbers and the flag h-vector of $J({\bf 2} \times {\bf n})$ | 13,747 |
We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in most cases generalize known results. | Generalized bivariate Fibonacci polynomials | 13,748 |
We present a method, illustrated by several examples, to find explicit counts of permutations containing a given multiset of three letter patterns. The method is recursive, depending on bijections to reduce to the case of a smaller multiset, and involves a consideration of separate cases according to how the patterns overlap. Specifically, we use the method (i) to provide combinatorial proofs of Bona's formula {2n-3}choose{n-3} for the number of n-permutations containing one 132 pattern and Noonan's formula 3/n {2n}choose{n+3} for one 123 pattern, (ii) to express the number of n-permutations containing exactly k 123 patterns in terms of ballot numbers for k<=4, and (iii) to express the number of 123-avoiding n-permutations containing exactly k 132 patterns as a linear combination of powers of 2, also for k<=4. The results strengthen the conjecture that the counts are algebraic for all k. | A recursive bijective approach to counting permutations containing
3-letter patterns | 13,749 |
We study some properties of the {\bf cd}-index of the Boolean lattice. They are extremely similar to the properties of the {\ab}-index, or equivalently, the flag $h$-vector of the Boolean lattice and hence may be viewed as their {\bf cd}-analogues. We define a different algebra structure on the polynomial algebra $k < \cv, \dv>$ and give a derivation on this algebra. It is of significance for the Boolean lattice and forms our main tool. Using similar methods, we also prove some results for the {\bf cd}-index of the cubical lattice. We show that the Dehn-Sommerville relations for the flag $f$-vector of an Eulerian poset are equivalent to certain simple identities that exist in our algebra. | The cd-index of the Boolean lattice | 13,750 |
We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z^2, and always stay in the quadrant x >= 0, y >= 0. We first give a criterion which guarantees that the length generating function of these walks is D-finite, that is, satisfies a linear differential equation with polynomial coefficients. This criterion applies, among others, to the ordinary square lattice walks. Then, we prove that walks that start from (1,1), take their steps in {(2,-1), (-1,2)} and stay in the first quadrant have a non-D-finite generating function. Our proof relies on a functional equation satisfied by this generating function, and on elementary complex analysis. | Walks confined in a quadrant are not always D-finite | 13,751 |
We propose a novel algorithm for enumerating and listing all minimal cutsets of a given graph. It is known that this problem is NP-hard. We use connectivity properties of a given graph to develop an algorithm with reduced complexity for finding all its cutsets. We use breadth first search (BFS) method in conjunction with edge contraction to develop the algorithm. We introduce the concepts of a pivot vertex and absorbable clusters and use them to develop an enhanced recursive contraction algorithm. The complexity of the proposed algorithm is proportionate to the number of cutsets. We present simulation results to compare the performance of our proposed algorithm with those of existing methods. | A novel and efficient algorithm for scanning all minimal cutsets of a
graph | 13,752 |
A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least 5 or the complement of one. The "strong perfect graph conjecture" (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornuejols and Vuskovic -- that every Berge graph either falls into one of a few basic classes, or it has a kind of separation that cannot occur in a minimal imperfect graph. In this paper we prove both these conjectures. | The strong perfect graph theorem | 13,753 |
Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cycle matroid of G is supersolvable iff G is chordal (rigid): this is another way to read Dirac's theorem on chordal graphs. Chordal binary matroids are not in general supersolvable. Nevertheless we prove that, for every supersolvable binary matroid M, a maximal chain of modular flats of M canonically determines a chordal graph. | How is a Chordal Graph like a Supersolvable Binary Matroid? | 13,754 |
Mitosis is a rule introduced by [Knutson-Miller, 2002] for manipulating subsets of the n by n grid. It provides an algorithm that lists the reduced pipe dreams (also known as rc-graphs) [Fomin-Kirillov, Bergeron-Billey] for a permutation w in S_n by downward induction on weak Bruhat order, thereby generating the coefficients of Schubert polynomials [Lascoux-Schutzenberger] inductively. This note provides a short and purely combinatorial proof of these properties of mitosis. | Mitosis recursion for coefficients of Schubert polynomials | 13,755 |
Bivariate generating functions for various subsets of the class of permutations containing no descending sequence of length three or more are determined. The notion of absolute indecomposability of a permutation is introduced, and used in enumerating permutations which have a block structure avoiding 321 and whose blocks also have such structure (recursively). Generalizations of these results are discussed. | The fine structure of 321 avoiding permutations | 13,756 |
In this paper we prove that among the permutations of length n with i fixed points and j excedances, the number of 321-avoiding ones equals the number of 132-avoiding ones, for all given i,j<=n. We use a new technique involving diagonals of non-rational generating functions. This theorem generalizes a recent result of Robertson, Saracino and Zeilberger, for which we also give another, more direct proof. | Fixed points and excedances in restricted permutations | 13,757 |
Starting from some considerations we make about the relations between certain difference statistics and the classical permutation statistics we study permutations whose inversion number and excedance difference coincide. It turns out that these (so-called bi-increasing) permutations are just the 321-avoiding ones. The paper investigates their excedance and descent structure. In particular, we find some nice combinatorial interpretations for the distribution coefficients of the number of excedances and descents, respectively, and their difference analogues over the bi-increasing permutations in terms of parallelogram polyominoes and 2-Motzkin paths. This yields a connection between restricted permutations, parallelogram polyominoes, and lattice paths that reveals the relations between several well-known bijections given for these objects (e.g. by Delest-Viennot, Billey-Jockusch-Stanley, Francon-Viennot, and Foata-Zeilberger). As an application, we enumerate skew diagrams according to their rank and give a simple combinatorial proof for a result concerning the symmetry of the joint distribution of the number of excedances and inversions, respectively, over the symmetric group. | The excedances and descents of bi-increasing permutations | 13,758 |
Define $I_n^k(\alpha)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $\alpha \in S_i$, for some $i \geq 2$, and define $I_n^k(\emptyset;\alpha)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which contain the pattern $\alpha \in S_i$, for some $i \geq 2$, exactly once. Let $i_n^k(\alpha)$ be the number of elements in $I_n^k(\alpha)$ and let $i_n^k(\emptyset;\alpha)$ be the number of elements in $I_n^k(\emptyset;\alpha)$. We investigate $I_n^k(\alpha)$ and $I_n^k(\emptyset;\alpha)$ for all $\alpha \in S_3$. In particular, we show that $i_n^k(132)=i_n^k(213)=i_n^k(321)$, $i_n^k(231)=i_n^k(312)$, $i_n^k(\emptyset;132) =i_n^k(\emptyset;213)$, and $i_n^k(\emptyset;231)=i_n^k(\emptyset;312)$ for all $0 \leq k \leq n$. | Refined Restricted Involutions | 13,759 |
We clarify the exposition of Phases 2 and 3a in "The Floyd-Warshall Algorithm, the AP and the TSP". We also improve and simplify theorem 3.6 . In line with clarifying the exposition, we change the matrices in examples 3.4 and 3.5 of "The Floyd-Warshall Algorithm, the AP and the TSP II". | The Floyd-Warshall Algorithm, the AP and the TSP III | 13,760 |
We present a bijection between 321- and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. This gives a simple combinatorial proof of recent results of Robertson, Saracino and Zeilberger, and the first author. We also show that our bijection preserves additional statistics, which extends the previous results. | Bijections for refined restricted permutations | 13,761 |
Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. For $n > k > d$ let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed. | The order of monochromatic subgraphs with a given minimum degree | 13,762 |
Combinatorics is a fundamental mathematical discipline as well as an essential component of many mathematical areas, and its study has experienced an impressive growth in recent years. One of the main reasons for this growth is the tight connection between Discrete Mathematics and Theoretical Computer Science, and the rapid development of the latter. While in the past many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, the modern theory has grown out of this early stage, and often relies on deep, well developed tools. This is a survey of two of the main general techniques that played a crucial role in the development of modern combinatorics; algebraic methods and probabilistic methods. Both will be illustrated by examples, focusing on the basic ideas and the connection to other areas. | Discrete mathematics: methods and challenges | 13,763 |
Let $\bar{P}$ be a sequence of length $2n$ in which each element of $\{1,2,...,n\}$ occurs twice. Let $P'$ be a closed curve in a closed surface $S$ having $n$ points of simple auto-intersections, inducing a 4-regular graph embedded in $S$ which is 2-face colorable. If the sequence of auto-intersections along $P'$ is given by $\bar{P}$, we say that is a {\em $P'$ 2-face colorable solution for the Gauss Code $\bar{P}$ on surface $S$} or a {\em lacet for $\bar{P}$ on $S$}. In this paper we present a necessary and sufficient condition yielding these solutions when $S$ is Klein bottle. The condition take the form of a system of $m$ linear equations in $2n$ variables over $\Z_2$, where $m \le n(n-1)/2$. Our solution generalize solutions for the projective plane and on the sphere. In a strong way, the Klein bottle is an extremal case admitting an affine linear solution: we show that the similar problem on the torus and on surfaces of higher connectivity are modelled by a quadratic system of equations. | An Affine Linear Solution for the 2-Face Colorable Gauss Code Problem in
the Klein Bottle and a Quadratic System for Arbitrary Closed Surfaces | 13,764 |
We consider the two permutation statistics which count the distinct pairs obtained from the last two terms of occurrences of patterns t_1...t_{m-2}m(m-1) and t_1...t_{m-2}(m-1)m in a permutation, respectively. By a simple involution in terms of permutation diagrams we will prove their equidistribution over the symmetric group. As special case we derive a one-to-one correspondence between permutations which avoid each of the patterns t_1...t_{m-2}m(m-1) in S_m and such ones which avoid each of the patterns t_1...t_{m-2}(m-1)m. For m=3, this correspondence coincides with the bijection given by Simion and Schmidt in their famous paper on restricted permutations. | A generalization of the Simion-Schmidt bijection for restricted
permutations | 13,765 |
Let A=(a_(ij)) be the generic n by n circulant matrix given by a_(ij)=x_(i+j), with subscripts on x interpreted mod n. Define d(n) (resp. p(n)) to be the number of terms in the determinant (resp. permanent) of A. The function p(n) is well-known and has several combinatorial interpretations. The function d(n), on the other hand, has not been studied previously. We show that when n is a prime power, d(n)=p(n). The proof uses symmetric functions. | The number of terms in the permanent and the determinant of a generic
circulant matrix | 13,766 |
If a graph $G_M$ is embedded into a closed surface $S$ such that $S \backslash G_M$ is a collection of disjoint open discs, then $M=(G_M,S)$ is called a {\em map}. A {\em zigzag} in a map $M$ is a closed path which alternates choosing, at each star of a vertex, the leftmost and the rightmost possibilities for its next edge. If a map has a single zigzag we show that the cyclic ordering of the edges along it induces linear transformations, $c_P$ and $c_{P^\sim}$ whose images and kernels are respectively the cycle and bond spaces (over GF(2)) of $G_M$ and $G_D$, where $D=(G_D,S)$ is the dual map of $M$. We prove that $Im(c_P \circ c_{P^\sim})$ is the intersection of the cycle spaces of $G_M$ and $G_D$, and that the dimension of this subspace is connectivity of $S$. Finally, if $M$ has also a single face, this face induces a linear transformation $c_D$ which is invertible: we show that $c_D^{-1} = c_{P^\sim}$. | On Maps with a Single Zigzag | 13,767 |
We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by ``explaining'' their zeros using an appropriate combinatorial extension of the objects under consideration to negative integer parameters. We apply this method to prove a new refinement of the Bender-Knuth (ex-)Conjecture, which easily implies the Bender-Knuth (ex-)Conjecture itself. This is probably the most elementary way to prove this result currently known. Furthermore we adapt our method to q-polynomials, which allows us to derive generating function results as well. Finally we use this method to give another proof for the enumeration of semistandard tableaux of a fixed shape, which is opposed to the Bender-Knuth (ex-)Conjecture refinement a multivariate application of our method. | A method for proving polynomial enumeration formulas | 13,768 |
Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. We prove that there exists a function $\hat c=\hat c(n)$ with $0<c<\hat c<C$ such that for any $\eps > 0$, as $n$ tends to infinity $$Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0$$ and $$Pr [ G(n,(1+\eps)\hat c/\sqrt{n}) \in \R ] \to 1.$$ A crucial tool that is used in the proof and is of independent interest is a generalization of Szemer\'edi's Regularity Lemma to a certain hypergraph setting. | A sharp threshold for random graphs with a monochromatic triangle in
every edge coloring | 13,769 |
In this paper we consider dimonoids, which are sets equipped with two associative binary operations. Dimonoids in the sense of J.-L. Loday are xamples of duplexes. The set of all permutations, gives an example of a duplex which is not a dimonoid. We construct a free duplex generated by a given set via planar trees and then we prove that the set of all permutations form a free duplex on an explicitly described set of generators. We also consider duplexes coming from planar binary trees and vertices of the cubes. We prove that these duplexes are free with one generator in appropriate variety of duplexes. | Sets with two associative operation | 13,770 |
Each labeled rooted tree is associated with a hyperplane arrangement, which is free with exponents given by the depths of the vertices of this tree. The intersection lattices of these arrangements are described through posets of forests. These posets are used to define coalgebras, whose dual algebras are shown to have a simple presentation by generators and relations. | Free hyperplane arrangements associated to labeled rooted trees | 13,771 |
We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation $\tau$ on k letters, or containing $\tau$ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind. | Restricted even permutations and Chebyshev polynomials | 13,772 |
Let $w_{n+2}=pw_{n+1}+qw_{n}$ for $n\geq0$ with $w_0=a$ and $w_1=b$. In this paper we find an explicit expression, in terms of determinants, for $\sum_{n\geq0} w_n^kx^n$ for any $k\geq1$. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results. | A note on sum of k-th power of Horadam's sequence | 13,773 |
For $S$ a set of positive integers, and $k$ and $r$ fixed positive integers, denote by $f(S,k;r)$ the least positive integer $n$ (if it exists) such that within every $r$-coloring of $\{1,2,...,n\}$ there must be a monochromatic sequence $\{x_{1},x_{2},...,x_{k}\}$ with $x_{i}-x_{i-1} \in S$ for $2 \leq i \leq k$. We consider the existence of $f(S,k;r)$ for various choices of $S$, as well as upper and lower bounds on this function. In particular, we show that this function exists for all $k$ if $S$ is an odd translate of the set of primes and $r=2$. | Avoiding Monochromatic Sequences With Special Gaps | 13,774 |
We give factorizations for weighted spanning tree enumerators of Cartesian products of complete graphs, keeping track of fine weights related to degree sequences and edge directions. Our methods combine Kirchhoff's Matrix-Tree Theorem with the technique of identification of factors. | Factorizations of some weighted spanning tree enumerators | 13,775 |
Let $A_n\subseteq S_n$ denote the alternating and the symmetric groups on $1,...,n$. MacMahaon's theorem, about the equi-distribution of the length and the major indices in $S_n$, has received far reaching refinements and generalizations, by Foata, Carlitz, Foata-Schutzenberger, Garsia-Gessel and followers. Our main goal is to find analogous statistics and identities for the alternating group $A_{n}$. A new statistic for $S_n$, {\it the delent number}, is introduced. This new statistic is involved with new $S_n$ equi-distribution identities, refining some of the results of Foata-Schutzenberger and Garsia-Gessel. By a certain covering map $f:A_{n+1}\to S_n$, such $S_n$ identities are `lifted' to $A_{n+1}$, yielding the corresponding $A_{n+1}$ equi-distribution identities. | Permutation Statistics on the Alternating Group | 13,776 |
We first prove that the set of domino tilings of a fixed finite figure is a distributive lattice, even in the case when the figure has holes. We then give a geometrical interpretation of the order given by this lattice, using (not necessarily local) transformations called {\em flips}. This study allows us to formulate an exhaustive generation algorithm and a uniform random sampling algorithm. We finally extend these results to other types of tilings (calisson tilings, tilings with bicolored Wang tiles). | Domino tilings and related models: space of configurations of domains
with holes | 13,777 |
Motivated by work of Stembridge, we study rank functions for Viennot's heaps of pieces. We produce a simple and sufficient criterion for a heap to be a ranked poset and apply the results to the heaps arising from fully commutative words in Coxeter groups. | On rank functions for heaps | 13,778 |
We prove a recent conjecture of Lassalle about positivity and integrality of coefficients in some polynomial expansions. We also give a combinatorial interpretation of those numbers. Finally, we show that this question is closely related to the fundamental problem of calculating the linearization coefficients for binomial coefficients. | Sur une generalisation des coefficients binomiaux | 13,779 |
We study the number of solutions of the general semigroup equation in one variable, $X^\al=X^\be$, as well as of the system of equations $X^2=X, Y^2=Y, XY=YX$ in $H\wr T_n$, the wreath product of an arbitrary finite group $H$ with the full transformation semigroup $T_n$ on $n$ letters. For these solution numbers, we provide explicit exact formulae, as well as asymptotic estimates. Our results concerning the first mentioned problem generalize earlier results by Harris and Schoenfeld (J. Combin. Theory Ser. A 3 (1967), 122-135) on the number of idempotents in $T_n$, and a partial result of Dress and the second author (Adv. in Math. 129 (1997), 188-221). Among the asymptotic tools employed are Hayman's method for the estimation of coefficients of analytic functions and the Poisson summation formula. | Equations in finite semigroups: Explicit enumeration and asymptotics of
solution numbers | 13,780 |
The ring of locally-constant integer-valued functions on the dominant chamber of the Shi arrangement is endowed with a filtration and a new basis, compatible with this filtration, is found. This basis is compared to the trivial basis. The ring is given a presentation by generators and relations. | Antichains of positive roots and Heaviside functions | 13,781 |
This paper discusses new analytic algorithms and software for the enumeration of all integer flows inside a network. Concrete applications abound in graph theory \cite{Jaeger}, representation theory \cite{kirillov}, and statistics \cite{persi}. Our methods clearly surpass traditional exhaustive enumeration and other algorithms and can even yield formulas when the input data contains some parameters. These methods are based on the study of rational functions with poles on arrangements of hyperplanes. | Counting Integer flows in Networks | 13,782 |
The Optical Transpose Interconnection System (OTIS) was proposed by Marsden et al. [Opt. Lett 18 (1993) 1083--1085] to implement very dense one-to-one interconnection between processors in a free space of optical interconnections. The system which allows one-to-one optical communications from p groups of q transmitters to q groups of p receivers, using electronic intragroup communications for each group of consecutive d processors, is denoted by OTIS(p,q,d). H(p,q,d) is the digraph which characterizes the underlying topology of the optical interconnection implemented by OTIS(p,q,d). A digraph has an OTIS(p,q,d) layout if it is isomorphic to H(p,q,d). Based on results of Coudert et al. [Networks 40 (2002) 155--164], we characterize all OTIS(p,q,d) layouts of De Bruijn digraph B(d,n) where both p and q are powers of d. Coudert et al. posed the conjecture that if B(d,n) has an OTIS(p,q,d) layout, then both p and q are powers of d. As an effort to prove this conjecture, we prove that H(p,q,d) is a line digraph if and only if both p and q are multiples of d. | OTIS Layouts of De Bruijn Digraphs | 13,783 |
We introduce the Orchard crossing number, which is defined in a similar way to the well-known rectilinear crossing number. We compute the Orchard crossing number for some simple families of graphs. We also prove some properties of this crossing number. Moreover, we define a variant of this crossing number which is tightly connected to the rectilinear crossing number, and compute it for some simple families of graphs. | The Orchard crossing number of an abstract graph | 13,784 |
A (partial) Latin square is a table of multiplication of a (partial) quasigroup. Multiplication of a (partial) quasigroup may be considered as a set of triples. We give a necessary and sufficient condition when a set of triples is a quotient of a (partial) Latin square. | Latin squares, partial latin squares and its generalized quotients | 13,785 |
In 1989, Vaughan Jones introduced spin models and showed that they could be used to form link invariants in two different ways--by constructing representations of the braid group, or by constructing partition functions. These spin models were subsequently generalized to so-called 4-weight spin models by Bannai and Bannai; these could be used to construct partition functions, but did not lead to braid group representations in any obvious way. Jaeger showed that spin models were intimately related to certain association schemes. Yamada gave a construction of a symmetric spin model on $4n$ vertices from each 4-weight spin model on $n$ vertices. In this paper we build on recent work with Munemasa to give a different proof to Yamada's result, and we analyse the structure of the association scheme attached to this spin model. | Bose-Mesner Algebras attached to Invertible Jones Pairs | 13,786 |
A technique called graphical condensation is used to prove various combinatorial identities among numbers of (perfect) matchings of planar bipartite graphs and tilings of regions. Graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph, and then re-partitioning the united matching (actually a multigraph) into matchings of two other subgraphs, in one of two possible ways. This technique can be used to enumerate perfect matchings of a wide variety of bipartite planar graphs. Applications include domino tilings of Aztec diamonds and rectangles, diabolo tilings of fortresses, plane partitions, and transpose complement plane partitions. | Applications of Graphical Condensation for Enumerating Matchings and
Tilings | 13,787 |
This is a revision of the paper archived previously on August 22, 2002. It corrects a mistake in Sec. 8 concerning eccentricities of graphs. From any given sequence of finite or infinite graphs, a nonstandard graph is constructed. The procedure is similar to an ultrapower construction on an internal set from a sequence of subsets of the real line, but now the individual entities are the vertices of the graphs instead of real numbers. The transfer principle is then invoked to extend several graph-theoretic results to the nonstandard case. After incidences and adjacencies between nonstandard vertices and edges are defined, several formulas regarding numbers of vertices and edges, and nonstandard versions of Eulerian graphs, Hamiltonian graphs, and a coloring theorem are established for these nonstandard graphs. Key Words: Nonstandard graphs, transfer principle, ultrapower constructions. | Nonstandard Graphs, Revised | 13,788 |
Inspired by the results of Baik, Deift and Johansson on the limiting distribution of the lengths of the longest increasing subsequences in random permutations, we find those limiting distributions for pattern-restricted permutations in which the pattern is any one of the six patterns of length 3. We show that the (132)-avoiding case is identical to the distribution of heights of ordered trees, and that the (321)-avoiding case has interesting connections with a well known theorem of Erd\H os-Szekeres. | Longest increasing subsequences in pattern-restricted permutations | 13,789 |
This paper is devoted to a study of mathematical structures arising from choice functions satisfying the path independence property (Plott functions). We broaden the notion of a choice function by allowing of empty choice. This enables us to define a lattice structure on the set of Plott functions. Moreover, this lattice is functorially dependent on its base. We introduce a natural convex structure on the set of linear orders (or words) and show that Plott functions are in one-to-one correspondence with convex subsets in this set of linear orders. That correspondence is compatible with both lattice structures. Keywords: Convex geometries, shuffle, linear orders, lattices, direct image, path independence, convex structure | Mathematics of Plott choice functions | 13,790 |
Let $a_{i,j}(n)$ denote the number of walks in $n$ steps from $(0,0)$ to $(i,j)$, with steps $(\pm 1,0)$ and $(0,\pm 1)$, never touching a point $(-k,0)$ with $k\ge 0$ after the starting point. \bous and Schaeffer conjectured a closed form for the number $a_{-i,i}(2n)$ when $i\ge 1$. In this paper, we prove their conjecture, and give a formula for $a_{-i,i}(2n)$ for $i\le -1$. | Proof of a Conjecture on the Slit Plane Problem | 13,791 |
We prove that a tournament with $n$ vertices has more than $0.13n^2(1+o(1))$ edge-disjoint transitive triples. We also prove some results on the existence of large packings of $k$-vertex transitive tournaments in an $n$-vertex tournament. Our proofs combine probabilistic arguments and some powerful packing results due to Wilson and to Frankl and R\"odl. | The number of edge disjoint transitive triples in a tournament | 13,792 |
As an instance of the B-polynomial, the circuit, or cycle, polynomial P(G(Gamma); w) of the generalized rooted product G(Gamma) of graphs was studied by Farrell and Rosenfeld ({\em Jour. Math. Sci. (India)}, 2000, \textbf{11}(1), 35--47) and Rosenfeld and Diudea ({\em Internet Electron. J. Mol. Des.}, 2002, \textbf{1}(3), 142--156). In both cases, the rooted product G(Gamma) was considered without any restrictions on graphs G and Gamma. Herein, we present a new general result and its corollaries concerning the case when the core graph G is restricted to be bipartite. The last instance of G(Gamma), as well as all its predecessors, can find chemical applications. | The Circuit Polynomial of the Restricted Rooted Product G(Gamma) of
Graphs with a Bipartite Core G | 13,793 |
Using the formalism of noncommutative symmetric functions, we derive the basic theory of the peak algebra of symmetric groups and of its graded Hopf dual. Our main result is to provide a representation theoretical interpretation of the peak algebra and its graded dual as Grothendieck rings of the tower of Hecke-Clifford algebras at $q=0$. | The peak algebra and the Hecke-Clifford algebras at $q=0$ | 13,794 |
An {\em antimagic labeling} of a graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called {\em antimagic} if it has an antimagic labeling. A conjecture of Ringel (see \cite{HaRi}) states that every connected graph, but $K_2$, is antimagic. Our main result validates this conjecture for graphs having minimum degree $\Omega(\log n)$. The proof combines probabilistic arguments with simple tools from analytic number theory and combinatorial techniques. We also prove that complete partite graphs (but $K_2$) and graphs with maximum degree at least $n-2$ are antimagic. | Dense graphs are antimagic | 13,795 |
A simple permutation is one which maps no proper non-singleton interval onto an interval. We consider the enumeration of simple permutations from several aspects. Our results include a straightforward relationship between the ordinary generating function for simple permutations and that for all permutations, that the coefficients of this series are not P-recursive, an asymptotic expansion for these coefficients, and a number of congruence results. | The enumeration of simple permutations | 13,796 |
We extend the classical coupon collector's problem to one in which two collectors are simultaneously and independently seeking collections of $d$ coupons. We find, in finite terms, the probability that the two collectors finish at the same trial, and we find, using the methods of Gessel-Viennot, the probability that the game has the following ``ballot-like'' character: the two collectors are tied with each other for some initial number of steps, and after that the player who first gains the lead remains ahead throughout the game. As a by-product we obtain the evaluation in finite terms of certain infinite series whose coefficients are powers and products of Stirling numbers of the second kind. We study the variant of the original coupon collector's problem in which a single collector wants to obtain at least $h$ copies of each coupon. Here we give a simpler derivation of results of Newman and Shepp, and extend those results. Finally we obtain the distribution of the number of coupons that have been obtained exactly once (``singletons'') at the conclusion of a successful coupon collecting sequence. | Some new aspects of the coupon-collector's problem | 13,797 |
Let $G=(V(G),E(G))$ be a planar digraph embedded in the plane in which all inner faces are equilateral triangles (with three edges in each), and let the union $\Rscr$ of these faces forms a convex polygon. The question is: given a function $\sigma$ on the boundary edges of $G$, does there exist a concave function $f$ on $\Rscr$ which is affinely linear within each bounded face and satisfies $f(v)-f(u)=\sigma(e)$ for each boundary edge $e=(u,v)$? The functions $\sigma$ admitting such an $f$ form a polyhedral cone $C$, and when the region $\Rscr$ is a triangle, $C$ turns out to be exactly the cone of boundary data of honeycombs. Studing honeycombs in connection with a problem on spectra of triples of zero-sum Hermitian matrices, Knutson, Tao, and Woodward \cite{KTW} showed that $C$ is described by linear inequalities of Horn's type with respect to so-called {\em puzzles}, along with obvious linear constraints. The purpose of this paper is to give an alternative proof of that result, working in terms of discrete concave finctions, rather than honeycombs, and using only linear programming and combinatorial tools. Moreover, we extend the result to an arbitrary convex polygon $\Rscr$. | Concave cocirculations in a triangular grid | 13,798 |
The maximum integer skew-symmetric flow problem (MSFP) generalizes both the maximum flow and maximum matching problems. It was introduced by Tutte in terms of self-conjugate flows in antisymmetrical digraphs. He showed that for these objects there are natural analogs of classical theoretical results on usual network flows, such as the flow decomposition, augmenting path, and max-flow min-cut theorems. We give unified and shorter proofs for those theoretical results. We then extend to MSFP the shortest augmenting path method of Edmonds and Karp and the blocking flow method of Dinits, obtaining algorithms with similar time bounds in general case. Moreover, in the cases of unit arc capacities and unit ``node capacities'' the blocking skew-symmetric flow algorithm has time bounds similar to those established in Even and Tarjan (1975) and Karzanov (1973) for Dinits' algorithm. In particular, this implies an algorithm for finding a maximum matching in a nonbipartite graph in $O(\sqrt{n}m)$ time, which matches the time bound for the algorithm of Micali and Vazirani. Finally, extending a clique compression technique of Feder and Motwani to particular skew-symmetric graphs, we speed up the implied maximum matching algorithm to run in $O(\sqrt{n}m\log(n^2/m)/\log{n})$ time, improving the best known bound for dense nonbipartite graphs. Also other theoretical and algorithmic results on skew-symmetric flows and their applications are presented. | Maximum Skew-Symmetric Flows and Matchings | 13,799 |
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