Bigraphical arrangements

Hopkins, Samuel B.; Perkinson, David

doi:10.1090/tran/6341

Cited by 18 publications

(27 citation statements)

References 11 publications

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“…Remark 5.1.3. The realizable part of the Bohne-Dress theorem states that the regular tilings of Z A by paralleletopes are dual to the generic perturbations of the central hyperplane arrangement defined by A. Hopkins and Perkinson [25] investigated generic bigraphical arrangements, i.e. generic perturbations of twice the graphical arrangement, and associated certain partial orientations, which they called admissible, to the regions in the complement of such an arrangement.…”

Section: Dilations the Ehrhart Polynomial And The Tutte Polynomialmentioning

confidence: 99%

Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory

Backman

Baker

Yuen

2019

Forum of Mathematics, Sigma

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Let M be a regular matroid. The Jacobian group Jac(M ) of M is a finite abelian group whose cardinality is equal to the number of bases of M . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) Jac(G) of a graph G (in which case bases of the corresponding regular matroid are spanning trees of G).There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph Jac(G) and spanning trees. However, most of the known bijections use vertices of G in some essential way and are inherently "non-matroidal". In this paper, we construct a family of explicit and easy-todescribe bijections between the Jacobian group of a regular matroid M and bases of M , many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of M admits a canonical simply transitive action on the set G(M ) of circuit-cocircuit reversal classes of M , and then define a family of combinatorial bijections β σ,σ * between G(M ) and bases of M . (Here σ (resp. σ * ) is an acyclic signature of the set of circuits (resp. cocircuits) of M .) We then give a geometric interpretation of each such map β = β σ,σ * in terms of zonotopal subdivisions which is used to verify that β is indeed a bijection.Finally, we give a combinatorial interpretation of lattice points in the zonotope Z; by passing to dilations we obtain a new derivation of Stanley's formula linking the Ehrhart polynomial of Z to the Tutte polynomial of M . arXiv:1701.01051v3 [math.CO] 23 Apr 20191 If e and e * are dual edges of G and G * , respectively, then given an orientation for e * we orient e by rotating the orientation of e * clockwise locally near the crossing of e and e * .

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Section: Dilations the Ehrhart Polynomial And The Tutte Polynomialmentioning

confidence: 99%

Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory

Backman

Baker

Yuen

2019

Forum of Mathematics, Sigma

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“…(1) It is natural to ask if we can extend our results to the entire class of truncated affine arrangements; or even to subarrangements thereof. The Shi(G) arrangements [3,7] and the G-Shi arrangements [20] have garnered plenty of attention in the recent past, and remain an active area of research. A rook-theoretic perspective on them might be of interest and lend new insight.…”

Section: Final Remarksmentioning

confidence: 99%

Gessel polynomials, rooks, and extended Linial arrangements

Tewari

2019

Journal of Combinatorial Theory, Series A

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We study a family of polynomials associated with ascent-descent statistics on labeled rooted plane k-ary trees introduced by Gessel, from a rook-theoretic perspective. We generalize the excedance statistic on permutations to maximal nonattacking rook placements on certain rectangular boards by decomposing them into boards of staircase shape. We then relate the number of maximal nonattacking rook placements on certain skew boards to the number of regions in extended Linial arrangements by establishing a relation between the factorial polynomial of those boards to the characteristic polynomial of extended Linial arrangements. Furthermore, we give a combinatorial interpretation of the number of bounded regions in extended Linial arrangements in the setting of labeled rooted plane k-ary trees. Finally, using the work of Goldman-Joichi-White, we identify graphs whose chromatic polynomials equal the characteristic polynomials of extended Linial arrangements upto a straightforward normalization.

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“…The valid pair associated with R is (w, I) where w = 5 2 1 7 6 9 3 4 8 ∈ S 9 and I = [1,4], [2,7], [4,9] , and we may represent it simply by 3 : 5 2 1 7 6 9 3 4 8 .…”

Section: The Number Of Ish-parking Functions With Reverse Center Of Amentioning

confidence: 99%

“…The notion of G-parking function was introduced by Postnikov and Shapiro in the construction of two algebras related to a general undirected graph G [11]. Later, Hopkins and Perkinson [7] showed that the labels of the Pak-Stanley labeling of the regions of a given hyperplane arrangement defined by G are exactly the G-parking functions, a fact that had been conjectured by Duval, Klivans and Martin [6]. Recently, Mazin [9] generalized this result to a very general class of hyperplane arrangements, with a similar concept based on a general directed multi -graph G. Whereas Hopkins and Perkinson's hyperplane arrangements include for example the (original) multidimensional Shi arrangement, Mazin's hyperplane arrangements include the multidimensional k-Shi arrangement, the multidimensional Ish arrangement and in fact all the arrangements we consider here.…”

Section: Introductionmentioning

confidence: 99%

Between Shi and Ish

Duarte

Oliveira

2018

Discrete Mathematics

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We introduce a new family of hyperplane arrangements in dimension n ≥ 3 that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of a given subfamily have the same number of regions -the connected components of the complement of the union of the hyperplanes -which can be bijectively labeled with the Pak-Stanley labeling. In addition, we show that, in the cases of the Shi and the Ish arrangements, the number of labels with reverse centers of a given length is equal, and conjecture that the same happens with all of the members of the family.

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Bigraphical arrangements

Cited by 18 publications

References 11 publications

Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory

Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory

Gessel polynomials, rooks, and extended Linial arrangements

Between Shi and Ish

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