Facial structures of lattice path matroid polytopes

An, Suhyung; Jung, JiYoon; Kim, Sang–Wook

doi:10.1016/j.disc.2019.111628

Cited by 5 publications

(15 citation statements)

References 20 publications

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“…In this section we define a class of full Higgs lift delta-matroids using lattice paths. This is a natural direction in which to extend the theory of lattice path matroids, which has proven to be a rich vein; for instance, see [1,3,5,16,17,19,21,22,25,26,28,29]. The concrete nature of the delta-matroids defined below may help readers get a better handle on delta-matroids, and it may suggest new avenues of investigation.…”

Section: Lattice Path Delta-matroidsmentioning

confidence: 98%

Delta-matroids as subsystems of sequences of Higgs lifts

Bonin

Chun

Noble

2021

Advances in Applied Mathematics

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In [30], Tardos studied special delta-matroids obtained from sequences of Higgs lifts; these are the full Higgs lift delta-matroids that we treat and around which all of our results revolve. We give an excluded-minor characterization of the class of full Higgs lift delta-matroids within the class of all delta-matroids, and we give similar characterizations of two other minor-closed classes of delta-matroids that we define using Higgs lifts. We introduce a minor-closed, dual-closed class of Higgs lift delta-matroids that arise from lattice paths. It follows from results of Bouchet that all delta-matroids can be obtained from full Higgs lift delta-matroids by removing certain feasible sets; to address which feasible sets can be removed, we give an excluded-minor characterization of deltamatroids within the more general structure of set systems. Many of these excluded minors occur again when we characterize the delta-matroids in which the collection of feasible sets is the union of the collections of bases of matroids of different ranks, and yet again when we require those matroids to have special properties, such as being paving.

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Section: Lattice Path Delta-matroidsmentioning

confidence: 98%

Delta-matroids as subsystems of sequences of Higgs lifts

Bonin

Chun

Noble

2021

Advances in Applied Mathematics

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“…We notice that the vertices of P S (1,2) correspond to the three bases u = st(U ) = (1, 1, 0), b = st(B) = (1, 0, 1) and l = st(L) = (0, 1, 1), see Figure 9. Figure 9.…”

Section: Distributive Polytopesmentioning

confidence: 99%

“…An LPM is called snake if it has at least two elements, it is connected and its diagram has no interior lattice points, see Figure 4. Note that snakes have also been called border strips in [1,2].…”

Section: It Is Known That Ifmentioning

confidence: 99%

“…The coefficients of h * P are the entries of the h * -vector of P . The function L P (t) can be expressed as (1) L P (t) = n j=0 h * j t + n − j n .…”

Section: Introductionmentioning

confidence: 99%

“…The class of lattice path matroids was first introduced by Bonin, de Mier, and Noy [5]. Many different aspects of lattice path matroids have been studied: excluded minor results [4], algebraic geometry notions [12,23,24], the Tutte polynomial [6,18,19], the associated basis polytope in connection with the combinatorics of Bergman complexes [12], its facial structure [1,2], specific decompositions in relation with Lafforgue's work [8] as well as the related cut-set expansion conjecture [9].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Lattice Path Matroid Polytopes: Integer Points and Ehrhart Polynomial

Knauer

Martínez-Sandoval

Alfonsín

2018

Discrete Comput Geom

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In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a characterization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove that lattice path matroid polytopes are affinely equivalent to a family of distributive polytopes. As applications we obtain two new infinite families of matroids verifying a conjecture of De Loera et. al. and present an explicit formula of the Ehrhart polynomial for one of them. arXiv:1701.05529v2 [math.CO]

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Lattice Path Matroids and Quotients

Benedetti-Velásquez,

Knauer

2024

Combinatorica

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We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as coefficients. Furthermore, we study full lattice path flag matroids and show that—contrary to arbitrary positroid flag matroids—they correspond to points in the nonnegative flag variety. At the basis of this result lies an identification of certain intervals of the strong Bruhat order with lattice path flag matroids. A recent conjecture of Mcalmon, Oh, and Xiang states a characterization of quotients of positroids. We use our results to prove this conjecture in the case of LPMs.

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Facial structures of lattice path matroid polytopes

Cited by 5 publications

References 20 publications

Delta-matroids as subsystems of sequences of Higgs lifts

Delta-matroids as subsystems of sequences of Higgs lifts

On Lattice Path Matroid Polytopes: Integer Points and Ehrhart Polynomial

Lattice Path Matroids and Quotients

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