Bijection Between Oriented Maps and Weighted Non-Oriented Maps

Czyżewska-Jankowska, Agnieszka; Śniady, Piotr

doi:10.37236/6718

Cited by 4 publications

(1 citation statement)

References 24 publications

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“…Sniady was able to find a bijection between these two index sets [Śni15a], which inspired us to investigate the combinatorial side of b-conjecture in the case of unicellular maps, presented in this paper. Note added in revision: After submission of the current paper, the aforementioned result ofŚniady appeared in a joint paper with Czyżewska-Jankowska [CJŚ17].…”

Section: Related Problemsmentioning

confidence: 99%

Top Degree Part in $b$-Conjecture for Unicellular Bipartite Maps

Dołęga¹

2017

Electron. J. Combin.

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Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; 1, 1 + β) with an additional parameter β that may be interpreted as a continuous deformation of the rooted bipartite maps generating series. Indeed, it has the property that for β ∈ {0, 1}, it specializes to the rooted, orientable (general, i.e. orientable or not, respectively) bipartite maps generating series. They made the following conjecture: coefficients of ψ are polynomials in β with positive integer coefficients that can be written as a multivariate generating series of rooted, general bipartite maps, where the exponent of β is an integer-valued statistic that in some sense "measures the non-orientability" of the corresponding bipartite map.We show that except for two special values of β = 0, 1 for which the combinatorial interpretation of the coefficients of ψ is known, there exists a third special value β = −1 for which the coefficients of ψ indexed by two partitions µ, ν, and one partition with only one part are given by rooted, orientable bipartite maps with arbitrary face degrees and black/white vertex degrees given by µ/ν, respectively. We show that this evaluation corresponds, up to a sign, to a top-degree part of the coefficients of ψ. As a consequence, we introduce a collection of integer-valued statistics of maps (η) such that the top-degree of the multivariate generating series of rooted bipartite maps with only one face (called unicellular) with respect to η gives the top-degree of the appropriate coefficients of ψ. Finally, we show that the b-conjecture holds true for all rooted, unicellular bipartite maps of genus at most 2.

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Section: Related Problemsmentioning

confidence: 99%

Top Degree Part in $b$-Conjecture for Unicellular Bipartite Maps

Dołęga¹

2017

Electron. J. Combin.

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Asymptotics of Jack characters

Śniady

2019

Journal of Combinatorial Theory, Series A

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Generating series of non-oriented constellations and marginal sums in the Matching-Jack conjecture

Dali¹

2022

Algebraic Combinatorics

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Using the description of hypermaps with matchings, Goulden and Jackson have given an expression of the generating series of rooted bipartite maps in terms of the zonal polynomials. We generalize this approach to the case of constellations on non-oriented surfaces that have recently been introduced by Chapuy and Dołęga. A key step in the proof is an encoding of constellations with tuples of matchings.We consider a one parameter deformation of the generating series of constellations using Jack polynomials and we introduce the coefficients c λ µ 0 ,...,µ k (b) obtained by the expansion of these functions in the power-sum basis. These coefficients are indexed by k +2 integer partitions and the deformation parameter b, and can be considered as a generalization for k > 1 of the connection coefficients introduced by Goulden and Jackson. We prove that when we take some marginal sums, these coefficients enumerate b-weighted k-tuples of matchings. This can be seen as an "disconnected" version of a recent result of Chapuy and Dołęga for constellations. For k = 1, this gives a partial answer to Goulden and Jackson Matching-Jack conjecture.Lassalle has formulated a positivity conjecture for the coefficients θ (α) µ (λ), defined as the coefficient of the Jack polynomial J (α) λ in the power-sum basis. We use the second main result of this paper to give a proof of this conjecture in the case of partitions λ with rectangular shape.

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Bijection Between Oriented Maps and Weighted Non-Oriented Maps

Cited by 4 publications

References 24 publications

Top Degree Part in $b$-Conjecture for Unicellular Bipartite Maps

Top Degree Part in $b$-Conjecture for Unicellular Bipartite Maps

Asymptotics of Jack characters

Generating series of non-oriented constellations and marginal sums in the Matching-Jack conjecture

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