Annals of Combinatorics最新文献

We study a new variant of connected coloring of graphs based on the concept of strong edge coloring (every color class forms an induced matching). In particular, an edge-colored path is strongly proper if its color sequence does not contain identical terms within a distance of at most two. A strong proper connected coloring of G is the one in which every pair of vertices is joined by at least one strongly proper path. Let ({{,textrm{spc},}}(G)) denote the least number of colors needed for such coloring of a graph G. We prove that the upper bound ({{,textrm{spc},}}(G)le 5) holds for any 2-connected graph G. On the other hand, we demonstrate that there are 2-connected graphs with arbitrarily large girth satisfying ({{,textrm{spc},}}(G)ge 4). Additionally, we prove that graphs whose cycle lengths are divisible by 3 satisfy ({{,textrm{spc},}}(G)le 3). We also consider briefly other connected colorings defined by various restrictions on color sequences of connecting paths. For instance, in a nonrepetitive connected coloring of G, every pair of vertices should be joined by a path whose color sequence is nonrepetitive, that is, it does not contain two adjacent identical blocks. We demonstrate that 2-connected graphs are 15-colorable, while 4-connected graphs are 6-colorable, in the connected nonrepetitive sense. A similar conclusion with a finite upper bound on the number of colors holds for a much wider variety of connected colorings corresponding to fairly general properties of sequences. We end the paper with some open problems of concrete and general nature.

Equivariant Ehrhart theory generalizes the study of lattice point enumeration to also account for the symmetries of a polytope under a linear group action. We present a catalogue of techniques with applications in this field, including zonotopal decompositions, symmetric triangulations, combinatorial interpretation of the (h^*)-polynomial, and certificates for the (non)existence of invariant nondegenerate hypersurfaces. We apply these methods to several families of examples including hypersimplices, orbit polytopes, and graphic zonotopes, expanding the library of polytopes for which their equivariant Ehrhart theory is known.

The Maclaurin inequalities for graphs are a broad generalisation of the classical theorems of Turán and Zykov. In a nutshell they provide an asymptotically sharp answer to the following question: what is the maximum number of cliques of size q in a graph with a given number of cliques of size s and a given clique number? We prove an extension of the graph Maclaurin inequalities with a weight function that captures the local structure of the graph. As a corollary, we settle a recent conjecture of Kirsch and Nir, which simultaneously encompass the previous localised results of Bradač, Malec and Tompkins and of Kirsch and Nir.

A classical parking function of length n is a list of positive integers ((a_1, a_2, ldots , a_n)) whose nondecreasing rearrangement (b_1 le b_2 le cdots le b_n) satisfies (b_i le i). The convex hull of all parking functions of length n is an n-dimensional polytope in ({mathbb {R}}^n), which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of ({textbf{x}})-parking functions for ({textbf{x}}=(a,b,dots ,b)), which we refer to as ({textbf{x}})-parking function polytopes. We explore connections between these ({textbf{x}})-parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of ({textbf{x}})-parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.

Let (S_k times S_{n-k}) be a maximal Young subgroup of the symmetric group (S_n). We introduce a basis ({{mathcal {B}}}_{n,k}) for the coset space (S_n/S_k times S_{n-k}) that is naturally parametrized by the set of standard Young tableaux with n boxes, at most two rows, and at most k boxes in the second row. The basis ({{mathcal {B}}}_{n,k}) has positivity properties that resemble those of a root system, and there is a composition series of the coset space in which each term is spanned by the basis elements that it contains. We prove that the spherical functions of the associated Gelfand pair are nonnegative linear combinations of the ({{mathcal {B}}}_{n,k}).

Petersen and Tenner defined the depth statistic for Coxeter group elements which, in the symmetric group, can be described in terms of a cost function on transpositions. We generalize that cost function to the other classical (finite and affine) Weyl groups, letting the cost of an individual reflection t be the distance between the integers transposed by t in the combinatorial representation of the group (à la Eriksson and Eriksson). Arbitrary group elements then have a well-defined cost, obtained by minimizing the sum of the transposition costs among all factorizations of the element. We show that the cost of arbitrary elements can be computed directly from the elements themselves using a simple, intrinsic formula.

A permutation statistic ({{,textrm{st},}}) is said to be shuffle-compatible if the distribution of ({{,textrm{st},}}) over the set of shuffles of two disjoint permutations (pi ) and (sigma ) depends only on ({{,textrm{st},}}pi ), ({{,textrm{st},}}sigma ), and the lengths of (pi ) and (sigma ). Shuffle-compatibility is implicit in Stanley’s early work on P-partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility centered around their notion of the shuffle algebra of a shuffle-compatible statistic. For a family of statistics called descent statistics, these shuffle algebras are isomorphic to quotients of the algebra of quasisymmetric functions. Recently, Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsema defined a version of shuffle-compatibility for statistics on cyclic permutations, and studied cyclic shuffle-compatibility through purely combinatorial means. In this paper, we define the cyclic shuffle algebra of a cyclic shuffle-compatible statistic, and develop an algebraic framework for cyclic shuffle-compatibility in which the role of quasisymmetric functions is replaced by the cyclic quasisymmetric functions recently introduced by Adin, Gessel, Reiner, and Roichman. We use our theory to provide explicit descriptions for the cyclic shuffle algebras of various cyclic permutation statistics, which in turn gives algebraic proofs for their cyclic shuffle-compatibility.

A permutation is called Grassmannian if it has at most one descent. The study of pattern avoidance in such permutations was initiated by Gil and Tomasko in 2021. We continue this work by studying Grassmannian permutations that avoid an increasing pattern. In particular, we count the Grassmannian permutations of size m avoiding the identity permutation of size k,  thus solving a conjecture made by Weiner. We also refine our counts to special classes such as odd Grassmannian permutations and Grassmannian involutions. We prove most of our results by relating Grassmannian permutations to Dyck paths and binary words.

To any simple graph (G), the clique graph operator (K) assigns the graph (K(G)), which is the intersection graph of the maximal complete subgraphs of (G). The iterated clique graphs are defined by (K^{0}(G)=G) and (K^{n}(G)=K(K^{n-1}(G))) for (nge 1). We associate topological concepts to graphs by means of the simplicial complex (textrm{Cl}(G)) of complete subgraphs of (G). Hence, we say that the graphs (G_{1}) and (G_{2}) are homotopic whenever (textrm{Cl}(G_{1})) and (textrm{Cl}(G_{2})) are. A graph (G) such that (K^{n}(G)simeq G) for all (nge 1) is called (K)-homotopy permanent. A graph is Helly if the collection of maximal complete subgraphs of (G) has the Helly property. Let (G) be a Helly graph. Escalante (1973) proved that (K(G)) is Helly, and Prisner (1992) proved that (Gsimeq K(G)), and so Helly graphs are (K)-homotopy permanent. We conjecture that if a graph (G) satisfies that (K^{m}(G)) is Helly for some (mge 1), then (G) is (K)-homotopy permanent. If a connected graph has maximum degree at most four and is different from the octahedral graph, we say that it is a low degree graph. It was recently proven that all low-degree graphs (G) satisfy that (K^{2}(G)) is Helly. In this paper, we show that all low-degree graphs have the homotopy type of a wedge or circumferences, and that they are (K)-homotopy permanent.