22,693 research outputs found

    Graph Orientations and Linear Extensions.

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    Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph, and therefore we can count the number of linear extensions of these posets. We want to know which choice of orientation maximizes the number of linear extensions of the corresponding poset, and this problem is solved essentially for comparability graphs and odd cycles, presenting several proofs. We then provide an inequality for general graphs and discuss further techniques

    Enumeration of PLCP-orientations of the 4-cube

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    The linear complementarity problem (LCP) provides a unified approach to many problems such as linear programs, convex quadratic programs, and bimatrix games. The general LCP is known to be NP-hard, but there are some promising results that suggest the possibility that the LCP with a P-matrix (PLCP) may be polynomial-time solvable. However, no polynomial-time algorithm for the PLCP has been found yet and the computational complexity of the PLCP remains open. Simple principal pivoting (SPP) algorithms, also known as Bard-type algorithms, are candidates for polynomial-time algorithms for the PLCP. In 1978, Stickney and Watson interpreted SPP algorithms as a family of algorithms that seek the sink of unique-sink orientations of nn-cubes. They performed the enumeration of the arising orientations of the 33-cube, hereafter called PLCP-orientations. In this paper, we present the enumeration of PLCP-orientations of the 44-cube.The enumeration is done via construction of oriented matroids generalizing P-matrices and realizability classification of oriented matroids.Some insights obtained in the computational experiments are presented as well

    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

    Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings

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    For a poset P, a subposet A, and an order preserving map F from A into the real numbers, the marked order polytope parametrizes the order preserving extensions of F to P. We show that the function counting integral-valued extensions is a piecewise polynomial in F and we prove a reciprocity statement in terms of order-reversing maps. We apply our results to give a geometric proof of a combinatorial reciprocity for monotone triangles due to Fischer and Riegler (2011) and we consider the enumerative problem of counting extensions of partial graph colorings of Herzberg and Murty (2007).Comment: 17 pages, 10 figures; V2: minor changes (including title); V3: examples included (suggested by referee), to appear in "SIAM Journal on Discrete Mathematics

    Morphisms and order ideals of toric posets

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    Toric posets are cyclic analogues of finite posets. They can be viewed combinatorially as equivalence classes of acyclic orientations generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper we study toric intervals, morphisms, and order ideals, and we provide a connection to cyclic reducibility and conjugacy in Coxeter groups.Comment: 28 pages, 8 figures. A 12-page "extended abstract" version appears as [v2
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