On the Enumeration of Plane Bipolar Posets and Transversal Structures

Fusy, Éric; Narmanli, Erkan; Schaeffer, Gilles

doi:10.1007/978-3-030-83823-2_90

Cited by 7 publications

(10 citation statements)

References 11 publications

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“…This belief is justified by Theorem 1.17 and the results in [BM20]. More precisely, -We conjecture that bipolar posets (introduced in [FNS21]) are in bijection with semi-Baxter permutations and converge to a γ-Liouville quantum sphere decorated with two SLE curves of angle θ(q), where γ and θ(q) are related to the parameters ρ, q appearing in Theorem 1.16 through the relations given in Theorem 1.17.…”

Section: Open Problemsmentioning

confidence: 71%

“…This result will explain directly at the continuum level various connections between decorated planar maps and constrained permutations investigated in the literature at the discrete level. More precisely, in the combinatorial literature, several bijections between families of constrained permutations and decorated planar maps have been investigated, see for instance [DGW96,Bón97,BBMF11,KSSU13,FNS21].…”

Section: The Skew Brownian Permuton In Liouville Quantum Gravity Theorymentioning

confidence: 99%

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The skew Brownian permuton: a new universality class for random constrained permutations

Borga¹

2021

Preprint

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We construct a new family of random permutons, called skew Brownian permuton, which describes the limits of various models of random constrained permutations. This family is parametrized by two real parameters.For a specific choice of the parameters, the skew Brownian permuton coincides with the Baxter permuton, i.e. the permuton limit of Baxter permutations. We prove that for another specific choice of the parameters, the skew Brownian permuton coincides with the biased Brownian separable permuton, a one-parameter family of permutons previously studied in the literature as limit of uniform permutations in substitution-closed classes. This brings two different limiting objects under the same roof, identifying a new larger universality class.The skew Brownian permuton is constructed in terms of flows of solutions of some stochastic differential equations (SDEs) driven by two-dimensional correlated Brownian excursions in the nonnegative quadrant. We call these SDEs skew perturbed Tanaka equations because they are a mixture of the perturbed Tanaka equations and the equations encoding skew Brownian motions. We prove existence and uniqueness of (strong) solutions for these new SDEs.In addition, we show that some natural permutons arising from SLE-decorated Liouville quantum spheres are skew Brownian permutons and such permutons cover almost the whole range of possible parameters. The connection between models of constrained permutations and models of decorated planar maps has been intensively investigated in the literature at the discrete level; this paper establishes this connection directly at the continuum level. Proving the latter result, we also give an SDE interpretation of some quantities related to SLE-decorated Liouville quantum spheres.Summarizing, this paper relates the following three families of random objects: permuton limits of constrained permutations, flows of solutions of skew perturbed Tanaka equations, and SLE-decorated Liouville quantum spheres.We finally collect a series of intriguing open-questions on these new limiting permutons.

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Section: Open Problemsmentioning

confidence: 71%

Section: The Skew Brownian Permuton In Liouville Quantum Gravity Theorymentioning

confidence: 99%

The skew Brownian permuton: a new universality class for random constrained permutations

Borga¹

2021

Preprint

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“…A system of two equations with two catalytic variables x, y can easily be written for the series Q e (t, x, y) and Q o (t, x, y) of P -admissible tandem walks staying in the quadrant, with even or odd final y positions, along the lines for instance of [2, Thm 3], and the same can be done for S-admissible tandem walks. The resulting equations are however somewhat cumbersome to manipulate and it turns out to be more efficient to reduce the problem to small step walk problems, in the spirit of [16,Prop. 4], but taking into account the final y parity.…”

Section: Enumerative Resultsmentioning

confidence: 99%

“…Similarly as in [16], once our models have been set in bijection to certain models of plane bipolar orientations, they can be set in bijection to specific quadrant walks by specializing a bijection due to Kenyon, Miller, Sheffield and Wilson (shortly called the KMSW bijection), which we use as a bijective black box. An example of the KMSW bijction is given on Figure 9.…”

Section: Bijections With Walks In the Quadrantmentioning

confidence: 99%

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Enumeration of corner polyhedra and 3-connected Schnyder labelings

Fusy¹,

Narmanli²,

Schaeffer³

2022

Preprint

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We show that corner polyhedra and 3-connected Schnyder labelings join the growing list of planar structures that can be set in exact correspondence with (weighted) models of quadrant walks via a bijection due to Kenyon, Miller, Sheffield and Wilson.Our approach leads to a first polynomial time algorithm to count these structures, and to the determination of their exact asymptotic growth constants: the number pn of corner polyhedra and sn of 3-connected Schnyder woods of size n respectively satisfy (pn) 1/n → 9/2 and (sn) 1/n → 16/3 as n goes to infinity.While the growth rates are rational, like in the case of previously known instances of such correspondences, the exponent of the asymptotic polynomial correction to the exponential growth does not appear to follow from the now standard Denisov-Wachtel approach, due to a bimodal behavior of the step set of the underlying tandem walk. However a heuristic argument suggests that these exponents are −1 − π/ arccos(9/16) ≈ −4.23 for pn and −1 − π/ arccos(22/27) ≈ −6.08 for sn, which would imply that the associated series are not D-finite.

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