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  • Lower bounds for graph bootstrap percolation via properties of polynomials
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-04-06
    Lianna Hambardzumyan; Hamed Hatami; Yingjie Qian

    We introduce a simple method for proving lower bounds for the size of the smallest percolating set in a certain graph bootstrap process. We apply this method to obtain recursive formulas for the sizes of the smallest percolating sets in multidimensional tori and multidimensional grids (in particular hypercubes). The former answers a question of Morrison and Noel [9], and the latter provides an alternative

    更新日期:2020-04-06
  • The canonical join complex of the Tamari lattice
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-04-06
    Emily Barnard

    In this paper, we study a simplicial complex on the elements of the Tamari lattice in types A and B called the canonical join complex. The canonical join representation of an element w in a lattice L is the unique lowest expression ⋁A for w, when such an expression exists. We say that the set A is a canonical join representation. The collection of all such subsets has the structure of an abstract simplicial

    更新日期:2020-04-06
  • Parametrizations of k-nonnegative matrices: Cluster algebras and k-positivity tests
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-04-02
    Anna Brosowsky; Sunita Chepuri; Alex Mason

    A k-positive matrix is a matrix where all minors of order k or less are positive. Computing all such minors to test for k-positivity is inefficient, as there are ∑ℓ=1k(nℓ)2 of them in an n×n matrix. However, there are minimal k-positivity tests which only require testing n2 minors. These minimal tests can be related by series of exchanges, and form a family of sub-cluster algebras of the cluster algebra

    更新日期:2020-04-03
  • On generalized Erdős–Ginzburg–Ziv constants for Z2d
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-03-31
    Alexander Sidorenko

    Let G be a finite abelian group, and r be a multiple of its exponent. The generalized Erdős–Ginzburg–Ziv constant sr(G) is the smallest integer s such that every sequence of length s over G has a zero-sum subsequence of length r. We find exact values of s2m(Z2d) for d≤2m+1. Connections to linear binary codes of maximal length and codes without a forbidden weight are discussed.

    更新日期:2020-03-31
  • Walks in the quarter plane: Genus zero case
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-03-31
    Thomas Dreyfus; Charlotte Hardouin; Julien Roques; Michael F. Singer

    We use Galois theory of difference equations to study the nature of the generating series of (weighted) walks in the quarter plane with genus zero kernel curve. Using this approach, we prove that the generating series do not satisfy any nontrivial (possibly nonlinear) algebraic differential equation with rational coefficients.

    更新日期:2020-03-31
  • Symmetric multisets of permutations
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-03-27
    Jonathan S. Bloom

    The following long-standing problem in combinatorics was first posed in 1993 by Gessel and Reutenauer [5]. For which multisubsets B of the symmetric group Sn is the quasisymmetric functionQ(B)=∑π∈BFDes(π),n a symmetric function? Here Des(π) is the descent set of π and FDes(π),n is Gessel's fundamental basis for the vector space of quasisymmetric functions. The purpose of this paper is to provide a

    更新日期:2020-03-27
  • Bijective proofs of skew Schur polynomial factorizations
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-03-26
    Arvind Ayyer; Ilse Fischer

    In a recent paper, Ayyer and Behrend present for a wide class of partitions factorizations of Schur polynomials with an even number of variables where half of the variables are the reciprocals of the others into symplectic and/or orthogonal group characters, thereby generalizing results of Ciucu and Krattenthaler for rectangular shapes. Their proofs proceed by manipulations of determinants underlying

    更新日期:2020-03-27
  • Partitioning the vertices of a torus into isomorphic subgraphs
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-03-26
    Marthe Bonamy; Natasha Morrison; Alex Scott

    Let H be an induced subgraph of the torus Ckm. We show that when k≥3 is even and |V(H)| divides some power of k, then for sufficiently large n the torus Ckn has a perfect vertex-packing with induced copies of H. On the other hand, disproving a conjecture of Gruslys, we show that when k is odd and not a prime power, then there exists H such that |V(H)| divides some power of k, but there is no n such

    更新日期:2020-03-27
  • The 1–2–3 Conjecture almost holds for regular graphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-03-25
    Jakub Przybyło

    The well-known 1–2–3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with 1, 2 and 3 so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be possible from the weight set {1,2,3,4,5}. We show that for regular graphs it is sufficient to use weights 1, 2, 3, 4. Moreover, we prove the conjecture to hold for every

    更新日期:2020-03-26
  • On a perfect matching in a random digraph with average out-degree below two
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-03-20
    Michal Karoński; Ed Overman; Boris Pittel

    Existence of a perfect matching in a random bipartite digraph with bipartition (V1,V2), |Vi|=n, is studied. The graph is generated in two rounds of random selections of a potential matching partner such that the average number of selections made by each vertex overall is below 2. More precisely, in the first round each vertex chooses a potential mate uniformly at random, and independently of all vertices

    更新日期:2020-03-21
  • Catalan intervals and uniquely sorted permutations
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-03-20
    Colin Defant

    For each positive integer k, we consider five well-studied posets defined on the set of Dyck paths of semilength k. We prove that uniquely sorted permutations avoiding various patterns are equinumerous with intervals in these posets. While most of our proofs are bijective, some use generating trees and generating functions. We end with several conjectures.

    更新日期:2020-03-21
  • The partition rank of a tensor and k-right corners in Fqn
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-03-17
    Eric Naslund

    Following the breakthrough of Croot, Lev, and Pach [4], Tao [10] introduced a symmetrized version of their argument, which is now known as the slice rank method. In this paper, we introduce a more general version of the slice rank of a tensor, which we call the Partition Rank. This allows us to extend the slice rank method to problems that require the variables to be distinct. Using the partition rank

    更新日期:2020-03-19
  • Exponential bounds for the Erdős-Ginzburg-Ziv constant
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-03-17
    Eric Naslund

    The Erdős-Ginzburg-Ziv constant of a finite abelian group G, denoted s(G), is the smallest k∈N such that any sequence of elements of G of length k contains a zero-sum subsequence of length exp⁡(G). In this paper, we use the partition rank from [14], which generalizes the slice rank, to prove that for any odd prime p,s(Fpn)≤(p−1)2p(J(p)⋅p)n where 0.84143, this is the

    更新日期:2020-03-19
  • Hedetniemi's conjecture is asymptotically false
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-03-13
    Xiaoyu He; Yuval Wigderson

    Extending a recent breakthrough of Shitov, we prove that the chromatic number of the tensor product of two graphs can be a constant factor smaller than the minimum chromatic number of the two graphs. More precisely, we prove that there exists an absolute constant δ>0 such that for all c sufficiently large, there exist graphs G and H with chromatic number at least (1+δ)c for which χ(G×H)≤c.

    更新日期:2020-03-16
  • Bi-pruned Hurwitz numbers
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-03-13
    Marvin Anas Hahn

    Hurwitz numbers enumerate ramified coverings of the Riemann sphere with fixed ramification data. Certain kinds of ramification data are of particular interest, such as double Hurwitz numbers, which count covers with fixed arbitrary ramification over 0 and ∞ and simple ramification over b points, where b is given by the Riemann-Hurwitz formula. In this work, we introduce the notion of bi-pruned double

    更新日期:2020-03-16
  • 2-factors with k cycles in Hamiltonian graphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-03-10
    Matija Bucić; Erik Jahn; Alexey Pokrovskiy; Benny Sudakov

    A well known generalisation of Dirac's theorem states that if a graph G on n≥4k vertices has minimum degree at least n/2 then G contains a 2-factor consisting of exactly k cycles. This is easily seen to be tight in terms of the bound on the minimum degree. However, if one assumes in addition that G is Hamiltonian it has been conjectured that the bound on the minimum degree may be relaxed. This was

    更新日期:2020-03-12
  • Two Erdős–Hajnal-type theorems in hypergraphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-03-10
    Michal Amir; Asaf Shapira; Mykhaylo Tyomkyn

    The Erdős–Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph H, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of r-uniform hypergraphs. A theorem of Rödl asserts that if an n-vertex graph is non-universal then it contains an

    更新日期:2020-03-10
  • Decompositions into isomorphic rainbow spanning trees
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-03-10
    Stefan Glock; Daniela Kühn; Richard Montgomery; Deryk Osthus

    A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. Our main result implies that, given any optimal colouring of a sufficiently large complete graph K2n, there exists a decomposition of K2n into isomorphic rainbow spanning trees. This settles conjectures of Brualdi–Hollingsworth (from 1996) and Constantine (from 2002) for large graphs.

    更新日期:2020-03-10
  • On k-connected-homogeneous graphs
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-03-09
    Alice Devillers; Joanna B. Fawcett; Cheryl E. Praeger; Jin-Xin Zhou

    A graph Γ is k-connected-homogeneous (k-CH) if k is a positive integer and any isomorphism between connected induced subgraphs of order at most k extends to an automorphism of Γ, and connected-homogeneous (CH) if this property holds for all k. Locally finite, locally connected graphs often fail to be 4-CH because of a combinatorial obstruction called the unique x property; we prove that this property

    更新日期:2020-03-09
  • Vector parking functions with periodic boundaries and rational parking functions
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-03-02
    Yue Cai; Catherine H. Yan

    Vector parking functions are sequences of non-negative integers whose order statistics are bounded by a given integer sequence u=(u0,u1,u2,…). Using the theory of fractional power series and an analog of Newton-Puiseux Theorem, we derive the exponential generating function for the number of u-parking functions when u is periodic. Our method is to convert an Appell relation of Gončarov polynomials to

    更新日期:2020-03-03
  • Linear representations of finite geometries and associated LDPC codes
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-03-02
    Peter Sin; Julien Sorci; Qing Xiang

    The linear representation of a subset of a finite projective space is an incidence system of affine points and lines determined by the subset. In this paper we use character theory to show that the rank of the incidence matrix has a direct geometric interpretation in terms of certain hyperplanes. We consider the LDPC codes defined by taking the incidence matrix and its transpose as parity-check matrices

    更新日期:2020-03-03
  • The Alon-Tarsi number of a planar graph minus a matching
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-03-02
    Jarosław Grytczuk; Xuding Zhu

    This paper proves that every planar graph G contains a matching M such that the Alon-Tarsi number of G−M is at most 4. As a consequence, G−M is 4-paintable, and hence G itself is 1-defective 4-paintable. This improves a result of Cushing and Kierstead (2010) [5], who proved that every planar graph is 1-defective 4-choosable.

    更新日期:2020-03-02
  • A superlinear lower bound on the number of 5-holes
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-02-27
    Oswin Aichholzer; Martin Balko; Thomas Hackl; Jan Kynčl; Irene Parada; Manfred Scheucher; Pavel Valtr; Birgit Vogtenhuber

    Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let h5(n) be the minimum number of 5-holes among all sets of n points in the plane in general position. Despite many efforts

    更新日期:2020-02-28
  • Stack sorting with restricted stacks
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-02-21
    Giulio Cerbai; Anders Claesson; Luca Ferrari

    The (classical) problem of characterizing and enumerating permutations that can be sorted using two stacks connected in series is still largely open. In the present paper we address a related problem, in which we impose restrictions both on the procedure and on the stacks. More precisely, we consider a greedy algorithm where we perform the rightmost legal operation. Moreover, the first stack is required

    更新日期:2020-02-21
  • A symplectic refinement of shifted Hecke insertion
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-02-21
    Eric Marberg

    Buch, Kresch, Shimozono, Tamvakis, and Yong defined Hecke insertion to formulate a combinatorial rule for the expansion of the stable Grothendieck polynomials Gπ indexed by permutations in the basis of stable Grothendieck polynomials Gλ indexed by partitions. Patrias and Pylyavskyy introduced a shifted analogue of Hecke insertion whose natural domain is the set of maximal chains in a weak order on

    更新日期:2020-02-21
  • Edge-critical subgraphs of Schrijver graphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-02-14
    Tomáš Kaiser; Matěj Stehlík

    For k≥1 and n≥2k, the Kneser graph KG(n,k) has all k-element subsets of an n-element set as vertices; two such subsets are adjacent if they are disjoint. It was first proved by Lovász that the chromatic number of KG(n,k) is n−2k+2. Schrijver constructed a vertex-critical subgraph SG(n,k) of KG(n,k) with the same chromatic number. For the stronger notion of criticality defined in terms of removing edges

    更新日期:2020-02-20
  • On the chromatic number of disjointness graphs of curves
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-02-14
    János Pach; István Tomon

    Let ω(G) and χ(G) denote the clique number and chromatic number of a graph G, respectively. The disjointness graph of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called x-monotone if every vertical line intersects it in at most one

    更新日期:2020-02-20
  • Recognizing sets of generators in finite polar spaces
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-02-12
    Stefaan De Winter; Jeroen Schillewaert

    We characterize non-singular polar spaces embedded in non-singular polar spaces of the same rank using subsets of generators satisfying a natural intersection condition. By [7] and [8] this result has applications to the theory of Cameron-Liebler sets. In fact, our result is a significant improvement of the main result in [7].

    更新日期:2020-02-20
  • Divergent permutations
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-02-12
    Emanuela Fachini; János Körner

    Two permutations of the natural numbers diverge if the absolute value of the difference of their elements in the same position goes to infinity. We show that there exists an infinite number of pairwise divergent permutations of the naturals. We relate this result to more general questions about the permutation capacity of infinite graphs.

    更新日期:2020-02-20
  • Characterization of queer supercrystals
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-02-14
    Maria Gillespie; Graham Hawkes; Wencin Poh; Anne Schilling

    We provide a characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al. This characterization is a combination of local queer axioms generalizing Stembridge's local axioms for crystal bases for simply-laced root systems, which were recently introduced by Assaf and Oguz, with further axioms and a new graph G characterizing the relations of the

    更新日期:2020-02-20
  • Degenerate Turán densities of sparse hypergraphs
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-02-13
    Chong Shangguan; Itzhak Tamo

    For fixed integers r>k≥2,e≥3, let fr(n,er−(e−1)k,e) be the maximum number of edges in an r-uniform hypergraph in which the union of any e distinct edges contains at least er−(e−1)k+1 vertices. A classical result of Brown, Erdős and Sós in 1973 showed that fr(n,er−(e−1)k,e)=Θ(nk). The degenerate Turán density is defined to be the limit (if it exists)π(r,k,e):=limn→∞⁡fr(n,er−(e−1)k,e)nk. Extending a

    更新日期:2020-02-20
  • On the Cheeger constant for distance-regular graphs
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-02-13
    Zhi Qiao; Jack H. Koolen; Greg Markowsky

    The Cheeger constant of a graph is the smallest possible ratio between the size of a subgraph and the size of its boundary. It is well known that this constant must be at least λ12, where λ1 is the smallest positive eigenvalue of the Laplacian matrix. The subject of this paper is a conjecture of the authors that for distance-regular graphs the Cheeger constant is at most λ1. In particular, we prove

    更新日期:2020-02-20
  • Families of lattice polytopes of mixed degree one
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-02-17
    Gabriele Balletti; Christopher Borger

    It has been shown by Soprunov that the normalized mixed volume (minus one) of an n-tuple of n-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined n-tuples of mixed degree at most one to be exactly those for which this lower bound is attained with equality, and posed the problem of a classification of such tuples

    更新日期:2020-02-20
  • A combinatorial approach for discrete car parking on random labelled trees
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-02-14
    Alois Panholzer

    We consider two analogues of a discrete version of the famous car parking problem of Rényi for trees, which are also known under the term random sequential adsorption. For both models, the blocking model, where cars arrive sequentially at the nodes and only park if the site and all neighbouring nodes are free, and the dimer model, where cars arrive sequentially at the edges and only park if both endnodes

    更新日期:2020-02-20
  • Recursions for rational q,t-Catalan numbers
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-02-17
    Eugene Gorsky; Mikhail Mazin; Monica Vazirani

    We give a simple recursion labeled by binary sequences which computes rational q,t-Catalan power series, both in relatively prime and non relatively prime cases. It is inspired by, but not identical to recursions due to B. Elias, M. Hogancamp, and A. Mellit, obtained in their study of link homology. We also compare our recursion with the Hogancamp-Mellit's recursion and verify a connection between

    更新日期:2020-02-20
  • The negative tetrahedron and the first infinite family of connected digraphs that are strongly determined by the Hermitian spectrum
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-02-20
    Pepijn Wissing; Edwin R. van Dam

    Thus far, digraphs that are uniquely determined by their Hermitian spectra have proven elusive. Instead, researchers have turned to spectral determination of classes of switching equivalent digraphs, rather than individual digraphs. In the present paper, we consider the traditional notion: a digraph (or mixed graph) is said to be strongly determined by its Hermitian spectrum (abbreviated SHDS) if it

    更新日期:2020-02-20
  • 3-Flows with large support
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-02-06
    Matt DeVos; Jessica McDonald; Irene Pivotto; Edita Rollová; Robert Šámal

    We prove that every 3-edge-connected graph G has a 3-flow ϕ with the property that |supp(ϕ)|≥56|E(G)|. The graph K4 demonstrates that this 56 ratio is best possible; there is an infinite family where 56 is tight.

    更新日期:2020-02-07
  • Induced subgraphs of graphs with large chromatic number. VI. Banana trees
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-01-27
    Alex Scott; Paul Seymour

    We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper [6], the first author proved that every tree has this property; and in another earlier paper with Maria Chudnovsky [2], we proved that every cycle has this property. Here we give a common

    更新日期:2020-01-27
  • Binomial Eulerian polynomials for colored permutations
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-27
    Christos A. Athanasiadis

    Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and Williams on the face enumeration of generalized permutohedra. They are γ-positive (in particular, palindromic and unimodal) polynomials which can be interpreted as h-polynomials of certain flag simplicial polytopes and which admit interesting Schur γ-positive symmetric function generalizations. This paper introduces analogues

    更新日期:2020-01-27
  • A combinatorial formula for the Ehrhart h⁎-vector of the hypersimplex
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-27
    Donghyun Kim

    We give a combinatorial formula for the Ehrhart h⁎-vector of the hypersimplex. In particular, we show that hd⁎(Δk,n) is the number of hypersimplicial decorated ordered set partitions of type (k,n) with winding number d, thereby proving a conjecture of N. Early. We do this by proving a more general conjecture of N. Early on the Ehrhart h⁎-vector of a generic cross-section of a hypercube.

    更新日期:2020-01-27
  • The hat guessing number of graphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-01-24
    Noga Alon; Omri Ben-Eliezer; Chong Shangguan; Itzhak Tamo

    Consider the following hat guessing game: n players are placed on n vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they

    更新日期:2020-01-24
  • Total non-negativity of some combinatorial matrices
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-23
    David Galvin; Adrian Pacurar

    Many combinatorial matrices — such as those of binomial coefficients, Stirling numbers of both kinds, and Lah numbers — are known to be totally non-negative, meaning that all minors (determinants of square submatrices) are non-negative. The examples noted above can be placed in a common framework: for each one there is a non-decreasing sequence (a1,a2,…), and a sequence (e1,e2,…), such that the (m

    更新日期:2020-01-24
  • Chromatic numbers of Kneser-type graphs
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-23
    Dmitriy Zakharov

    Let G(n,r,s) be a graph whose vertices are all r-element subsets of an n-element set, in which two vertices are adjacent if they intersect in exactly s elements. In this paper we study chromatic numbers of G(n,r,s) with r,s being fixed constants and n tending to infinity. Using a recent result of Keevash on existence of designs we deduce an inequality χ(G(n,r,s))⩽(1+o(1))nr−s(r−s−1)!(2r−2s−1)! for

    更新日期:2020-01-24
  • Rainbow factors in hypergraphs
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-23
    Matthew Coulson; Peter Keevash; Guillem Perarnau; Liana Yepremyan

    For any r-graph H, we consider the problem of finding a rainbow H-factor in an r-graph G with large minimum ℓ-degree and an edge-colouring that is suitably bounded. We show that the asymptotic degree threshold is the same as that for finding an H-factor.

    更新日期:2020-01-24
  • Maximum number of colourings: 4-chromatic graphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-01-23
    Fiachra Knox; Bojan Mohar

    It is proved that every connected graph G on n vertices with χ(G)≥4 has at most k(k−1)n−3(k−2)(k−3) k-colourings for every k≥4. Equality holds for some (and then for every) k if and only if the graph is formed from K4 by repeatedly adding leaves. This confirms (a strengthening of) the 4-chromatic case of a long-standing conjecture of Tomescu [29]. Proof methods may be of independent interest. In particular

    更新日期:2020-01-23
  • e-Positivity of vertical strip LLT polynomials
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-22
    Michele D'Adderio

    In this article we prove the e-positivity of Gν[X;q+1] when Gν[X;q] is a vertical strip LLT polynomial. This property has been conjectured in [2] and [7], and it implies several e-positivities conjectured in those references and in [3]. We make use of a result of Carlsson and Mellit [5] that shows that a vertical strip LLT polynomial can be obtained by applying certain compositions of operators of

    更新日期:2020-01-23
  • Counting 3-stack-sortable permutations
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-21
    Colin Defant

    We prove a “decomposition lemma” that allows us to count preimages of certain sets of permutations under West's stack-sorting map s. As a first application, we give a new proof of Zeilberger's formula for the number W2(n) of 2-stack-sortable permutations in Sn. Our proof generalizes, allowing us to find an algebraic equation satisfied by the generating function that counts 2-stack-sortable permutations

    更新日期:2020-01-22
  • The Steep-Bounce zeta map in Parabolic Cataland
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-20
    Cesar Ceballos; Wenjie Fang; Henri Mühle

    As a classical object, the Tamari lattice has many generalizations, including ν-Tamari lattices and parabolic Tamari lattices. In this article, we unify these generalizations in a bijective fashion. We first prove that parabolic Tamari lattices are isomorphic to ν-Tamari lattices for bounce paths ν. We then introduce a new combinatorial object called “left-aligned colorable tree”, and show that it

    更新日期:2020-01-21
  • Elementary moves on lattice polytopes
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-16
    Julien David; Lionel Pournin; Rado Rakotonarivo

    We introduce a graph structure on the set of Euclidean polytopes. The vertices of this graph are the d-dimensional polytopes contained in Rd and its edges connect any two polytopes that can be obtained from one another by either inserting or deleting a vertex, while keeping their vertex sets otherwise unaffected. We prove several results on the connectivity of this graph, and on a number of its subgraphs

    更新日期:2020-01-21
  • Maximum degree and diversity in intersecting hypergraphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-01-15
    Peter Frankl

    Let S be an n-element set and F⊂(Sk) an intersecting family. Improving earlier results it is proved that for n>72k there is an element of S that is contained in all but (n−3k−2) members of F. One of the main ingredients of the proof is the following statement. If G⊂(Sk) is intersecting, |G|≥(n−2k−2) and n≥72k then there is an element of S that is contained in more than half of the members of G.

    更新日期:2020-01-16
  • The EKR property for flag pure simplicial complexes without boundary
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-14
    Jorge Alberto Olarte; Francisco Santos; Jonathan Spreer; Christian Stump

    We prove that the family of facets of a pure simplicial complex C of dimension up to three satisfies the Erdős-Ko-Rado property whenever C is flag and has no boundary ridges. We conjecture the same to be true in arbitrary dimension and give evidence for this conjecture. Our motivation is that complexes with these two properties include flag pseudo-manifolds and cluster complexes.

    更新日期:2020-01-15
  • Chordality, d-collapsibility, and componentwise linear ideals
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-14
    Mina Bigdeli; Sara Faridi

    Using the concept of d-collapsibility from combinatorial topology, we define chordal simplicial complexes and show that their Stanley-Reisner ideals are componentwise linear. Our construction is inspired by and an extension of “chordal clutters” which was defined by Bigdeli, Yazdan Pour and Zaare-Nahandi in 2017, and characterizes Betti tables of all ideals with a linear resolution in a polynomial

    更新日期:2020-01-15
  • On the ℓ4:ℓ2 ratio of functions with restricted Fourier support
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-14
    Naomi Kirshner; Alex Samorodnitsky

    Given a subset A⊆{0,1}n, let μ(A) be the maximal ratio between ℓ4 and ℓ2 norms of a function whose Fourier support is a subset of A.1 We make some simple observations about the connections between μ(A) and the additive properties of A on one hand, and between μ(A) and the uncertainty principle for A on the other hand. One application obtained by combining these observations with results in additive

    更新日期:2020-01-15
  • On the interplay between additive and multiplicative largeness and its combinatorial applications
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-13
    Vitaly Bergelson; Daniel Glasscock

    Many natural notions of additive and multiplicative largeness arise from results in Ramsey theory. In this paper, we explain the relationships between these notions for subsets of N and in more general ring-theoretic structures. We show that multiplicative largeness begets additive largeness in three ways and give a collection of examples demonstrating the optimality of these results. We also give

    更新日期:2020-01-14
  • Hamiltonicity in randomly perturbed hypergraphs
    J. Comb. Theory B (IF 0.892) Pub Date : 2020-01-10
    Jie Han; Yi Zhao

    For integers k≥3 and 1≤ℓ≤k−1, we prove that for any α>0, there exist ϵ>0 and C>0 such that for sufficiently large n∈(k−ℓ)N, the union of a k-uniform hypergraph with minimum vertex degree αnk−1 and a binomial random k-uniform hypergraph G(k)(n,p) with p≥n−(k−ℓ)−ϵ for ℓ≥2 and p≥Cn−(k−1) for ℓ=1 on the same vertex set contains a Hamiltonian ℓ-cycle with high probability. Our result is best possible up

    更新日期:2020-01-11
  • Toeplitz minors and specializations of skew Schur polynomials
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-06
    David García-García; Miguel Tierz

    We express minors of Toeplitz matrices of finite and large dimension in terms of symmetric functions. Comparing the resulting expressions with the inverses of some Toeplitz matrices, we obtain explicit formulas for a Selberg-Morris integral and for specializations of certain skew Schur polynomials.

    更新日期:2020-01-07
  • 3D positive lattice walks and spherical triangles
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-06
    B. Bogosel; V. Perrollaz; K. Raschel; A. Trotignon

    In this paper we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. As shown in [29], both the exponential growth and the critical exponent admit universal formulas, respectively in terms of the inventory of the step set and of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. We

    更新日期:2020-01-07
  • New necessary conditions on (negative) Latin square type partial difference sets in abelian groups
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-07
    Zeying Wang

    Partial difference sets (for short, PDSs) with parameters (n2, r(n−ϵ), ϵn+r2−3ϵr, r2−ϵr) are called Latin square type (respectively negative Latin square type) PDSs if ϵ=1 (respectively ϵ=−1). In this paper, we will give restrictions on the parameter r of a (negative) Latin square type partial difference set in an abelian group of non-prime power order a2b2, where gcd⁡(a,b)=1, a>1, and b is an odd

    更新日期:2020-01-07
  • A generalized Eulerian triangle from staircase tableaux and tree-like tableaux
    J. Comb. Theory A (IF 0.958) Pub Date : 2020-01-07
    Bao-Xuan Zhu

    Motivated by the classical Eulerian triangle and triangular arrays from staircase tableaux and tree-like tableaux, we study a generalized Eulerian array [Tn,k]n,k≥0, which satisfies the recurrence relation:Tn,k=λ(a1k+a2)Tn−1,k+[(b1−da1)n−(b1−2da1)k+b2−d(a1−a2)]Tn−1,k−1+d(b1−da1)λ(n−k+1)Tn−1,k−2, where T0,0=1 and Tn,k=0 unless 0≤k≤n. We derive some properties of [Tn,k]n,k≥0, including the explicit formulae

    更新日期:2020-01-07
  • On cubic graphical regular representations of finite simple groups
    J. Comb. Theory B (IF 0.892) Pub Date : 2019-06-13
    Binzhou Xia

    A recent conjecture of the author and Teng Fang states that there are only finitely many finite simple groups with no cubic graphical regular representation. In this paper, we make crucial progress towards this conjecture by giving an affirmative answer for groups of Lie type of large rank.

    更新日期:2020-01-04
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