25,154 research outputs found

    Brick polytopes, lattice quotients, and Hopf algebras

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    This paper is motivated by the interplay between the Tamari lattice, J.-L. Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf algebra on binary trees. We show that these constructions extend in the world of acyclic kk-triangulations, which were already considered as the vertices of V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural surjection from the permutations to the acyclic kk-triangulations. We show that the fibers of this surjection are the classes of the congruence k\equiv^k on Sn\mathfrak{S}_n defined as the transitive closure of the rewriting rule UacV1b1VkbkWkUcaV1b1VkbkWU ac V_1 b_1 \cdots V_k b_k W \equiv^k U ca V_1 b_1 \cdots V_k b_k W for letters a<b1,,bk<ca < b_1, \dots, b_k < c and words U,V1,,Vk,WU, V_1, \dots, V_k, W on [n][n]. We then show that the increasing flip order on kk-triangulations is the lattice quotient of the weak order by this congruence. Moreover, we use this surjection to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on permutations, indexed by acyclic kk-triangulations, and to describe the product and coproduct in this algebra and its dual in term of combinatorial operations on acyclic kk-triangulations. Finally, we extend our results in three directions, describing a Cambrian, a tuple, and a Schr\"oder version of these constructions.Comment: 59 pages, 32 figure

    Brick assignments and homogeneously almost self-complementary graphs

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    AbstractA graph is called almost self-complementary if it is isomorphic to the graph obtained from its complement by removing a 1-factor. In this paper, we study a special class of vertex-transitive almost self-complementary graphs called homogeneously almost self-complementary. These graphs occur as factors of symmetric index-2 homogeneous factorizations of the “cocktail party graphs” K2n−nK2. We construct several infinite families of homogeneously almost self-complementary graphs, study their structure, and prove several classification results, including the characterization of all integers n of the form n=pr and n=2p with p prime for which there exists a homogeneously almost self-complementary graph on 2n vertices

    Sharing Stories as Legacy: What Matters to Older Adults?

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    Objectives: Legacy allows individuals to make meaning of their lives by passing on their experiences and beliefs to younger people and influencing their perspectives, perceptions, and actions. This mixed-methods study investigated: (1) What is important for older adults to share as legacy with families, friends and others, based on the types and features of their digital stories ? and (2) How do older adults’ digital stories affect story viewers? Methods: One hundred adults aged between 55 and 95 years participated in ten-week Elder’s Digital Storytelling courses and created short digital stories. Using the content analysis approach, the story transcripts were thematically analyzed and iteratively coded by three researchers and the results were quantified. A diverse group of 60 viewers at a public event provided their reactions to the digital stories. Results: The findings revealed that character, place, and family were chosen as the primary types by the older adults for their legacy digital stories. Accomplishment and career/school were the next most prominent story types. Moreover, these digital stories appeared to have a powerful impact on the viewers. Discussion: A digital story is a powerful artifact to communicate an older person’s legacy because it is based on familiar forms of communication, such as speech and photographs. The major legacy themes chosen by the older adults align with the findings of the research literature. The feedback from the viewers of the digital stories reflects these as a source of life wisdom and legacy for younger generations. Funding details: This work was supported by the AGE-WELL National Centre of Excellence (AW CRP 2015-WP4.3)

    The brick polytope of a sorting network

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    The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of flip graphs on pseudoline arrangements with contacts supported by a given sorting network. In this paper, we construct the brick polytope of a sorting network, obtained as the convex hull of the brick vectors associated to each pseudoline arrangement supported by the network. We combinatorially characterize the vertices of this polytope, describe its faces, and decompose it as a Minkowski sum of matroid polytopes. Our brick polytopes include Hohlweg and Lange's many realizations of the associahedron, which arise as brick polytopes for certain well-chosen sorting networks. We furthermore discuss the brick polytopes of sorting networks supporting pseudoline arrangements which correspond to multitriangulations of convex polygons: our polytopes only realize subgraphs of the flip graphs on multitriangulations and they cannot appear as projections of a hypothetical multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization of our results to spherical subword complexes on finite Coxeter groups (http://arxiv.org/abs/1111.3349

    A structure theorem for product sets in extra special groups

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    Hegyv\'ari and Hennecart showed that if BB is a sufficiently large brick of a Heisenberg group, then the product set BBB\cdot B contains many cosets of the center of the group. We give a new, robust proof of this theorem that extends to all extra special groups as well as to a large family of quasigroups.Comment: This manuscript has been updated to include referee corrections. To appear in Journal of Number Theor

    Rainbow Connection Number in the Brick Product Graphs C(2n,m, r)

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    All graphs considered in this paper are simple, finite and undirected. Let G be a nontrivial connected graph with an edge coloring c : E(G) → {1, 2, · · · , k}, k ∈ N, where adjacent edges may be colored the same. A path in G is called a rainbow path if no two edges of it are colored the same

    Snarks with total chromatic number 5

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    A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by chi(T)(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with chi(T) = 4 are said to be Type 1, and cubic graphs with chi(T) = 5 are said to be Type 2. Snarks are cyclically 4-edge-connected cubic graphs that do not allow a 3-edge-coloring. In 2003, Cavicchioli et al. asked for a Type 2 snark with girth at least 5. As neither Type 2 cubic graphs with girth at least 5 nor Type 2 snarks are known, this is taking two steps at once, and the two requirements of being a snark and having girth at least 5 should better be treated independently. In this paper we will show that the property of being a snark can be combined with being Type 2. We will give a construction that gives Type 2 snarks for each even vertex number n >= 40. We will also give the result of a computer search showing that among all Type 2 cubic graphs on up to 32 vertices, all but three contain an induced chordless cycle of length 4. These three exceptions contain triangles. The question of the existence of a Type 2 cubic graph with girth at least 5 remains open
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