Diagonally convex directed polyominoes and even trees: a bijection and related issues

Deutsch, Emeric; Feretić, Svjetlan; Noy, Marc

doi:10.1016/s0012-365x(02)00340-0

Cited by 19 publications

(22 citation statements)

References 12 publications

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“…Indeed, if this were not the case, for every j ∈ ((z − 2η)n, z n ), we would have W so that y n p n < r n < j n z n < l n and W (n) i = m n for every p n < i < r n . For the first inequality, note that p n < y n would imply W (n) y n m n and so min{i > y n : W (n) i W (n) y n } z n which, by (12), contradicts (13). In addition, for every n sufficiently large, s n −1 p n , n −1 j n < s + 2ε, n −1 j n − n −1 p n 2ε, t − 2ε < n −1 l n < t.…”

Section: Lemma 15mentioning

confidence: 99%

Triangulating stable laminations

Kortchemski¹,

Marzouk²

2016

Electron. J. Probab.

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We study the asymptotic behavior of random simply generated noncrossing planar trees in the space of compact subsets of the unit disk, equipped with the Hausdorff distance. Their distributional limits are obtained by triangulating at random the faces of stable laminations, which are random compact subsets of the unit disk made of non-intersecting chords coded by stable L\'evy processes. We also study other ways to "fill-in" the faces of stable laminations, which leads us to introduce the iteration of laminations and of trees.Comment: 34 pages, 5 figure

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Section: Lemma 15mentioning

confidence: 99%

Triangulating stable laminations

Kortchemski¹,

Marzouk²

2016

Electron. J. Probab.

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“…Besides taking the dual of a non-separable map, there are several other involutions on combinatorial objects whose fixed points are known to be equinumerous with fixed points of h. We first review some results from [3], where three classes of trees and a class of polyominoes are counted under reflection, and later we point at another connection with non-separable maps.…”

Section: More Open Bijective Questionsmentioning

confidence: 99%

Enumeration of Fixed Points of an Involution on β(1, 0)-Trees

Kitaev

Mier

2013

Graphs and Combinatorics

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β(1, 0)-trees provide a convenient description of rooted non-separable planar maps. The involution h on β(1, 0)-trees was introduced to prove a complicated equidistribution result on a class of pattern-avoiding permutations. In this paper, we describe and enumerate fixed points of the involution h. Intriguingly, the fixed points are equinumerous with the fixed points under taking the dual map on rooted non-separable planar maps, even though the fixed points do not go to each other under the know (natural) bijection between the trees and the maps.

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“…The vertex 1 will be also called the root, and we shall depict it as a top vertex. Non-crossing trees have been studied by Chen et al [2], Deutsch et al [3,4], Flajolet et al [5], Hough [6], Noy et al [7], and Panholzer et al [9]. Denote the set of all NC-trees on [n] by N C(n).…”

Section: Non-crossing Treesmentioning

confidence: 99%