172 research outputs found
Baxter permutations and plane bipolar orientations
We present a simple bijection between Baxter permutations of size and
plane bipolar orientations with n edges. This bijection translates several
classical parameters of permutations (number of ascents, right-to-left maxima,
left-to-right minima...) into natural parameters of plane bipolar orientations
(number of vertices, degree of the sink, degree of the source...), and has
remarkable symmetry properties. By specializing it to Baxter permutations
avoiding the pattern 2413, we obtain a bijection with non-separable planar
maps. A further specialization yields a bijection between permutations avoiding
2413 and 3142 and series-parallel maps.Comment: 22 page
Bijections for Baxter Families and Related Objects
The Baxter number can be written as . These
numbers have first appeared in the enumeration of so-called Baxter
permutations; is the number of Baxter permutations of size , and
is the number of Baxter permutations with descents and
rises. With a series of bijections we identify several families of
combinatorial objects counted by the numbers . Apart from Baxter
permutations, these include plane bipolar orientations with vertices and
faces, 2-orientations of planar quadrangulations with white and
black vertices, certain pairs of binary trees with left and
right leaves, and a family of triples of non-intersecting lattice paths. This
last family allows us to determine the value of as an
application of the lemma of Gessel and Viennot. The approach also allows us to
count certain other subfamilies, e.g., alternating Baxter permutations, objects
with symmetries and, via a bijection with a class of plan bipolar orientations
also Schnyder woods of triangulations, which are known to be in bijection with
3-orientations.Comment: 31 pages, 22 figures, submitted to JCT
Scaling and Local Limits of Baxter Permutations Through Coalescent-Walk Processes
Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called coalescent-walk processes, that are fundamental for our results. We relate these new objects with the other previously mentioned families introducing some new bijections.
We prove joint Benjamini - Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new fractal random measure of the unit square, called the coalescent Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations.
To prove the latter result, we study the scaling limit of the associated random coalescent-walk processes. We show that they converge in law to a continuous random coalescent-walk process encoded by a perturbed version of the Tanaka stochastic differential equation. This result has connections (to be explored in future projects) with the results of Gwynne, Holden, Sun (2016) on scaling limits (in the Peanosphere topology) of plane bipolar triangulations.
We further prove some results that relate the limiting objects of the four families to each other, both in the local and scaling limit case
Restricted non-separable planar maps and some pattern avoiding permutations
Tutte founded the theory of enumeration of planar maps in a series of papers
in the 1960s. Rooted non-separable planar maps are in bijection with
West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori,
Jacquard and Schaeffer in 1997, as well as a family of permutations defined by
the avoidance of two four letter patterns. In this paper we give upper and
lower bounds on the number of multiple-edge-free rooted non-separable planar
maps. We also use the bijection between rooted non-separable planar maps and a
certain class of permutations, found by Claesson, Kitaev and Steingrimsson in
2009, to show that the number of 2-faces (excluding the root-face) in a map
equals the number of occurrences of a certain mesh pattern in the permutations.
We further show that this number is also the number of nodes in the
corresponding beta(1,0)-tree that are single children with maximum label.
Finally, we give asymptotics for some of our enumerative results.Comment: 18 pages, 14 figure
Generic method for bijections between blossoming trees and planar maps
This article presents a unified bijective scheme between planar maps and
blossoming trees, where a blossoming tree is defined as a spanning tree of the
map decorated with some dangling half-edges that enable to reconstruct its
faces. Our method generalizes a previous construction of Bernardi by loosening
its conditions of applications so as to include annular maps, that is maps
embedded in the plane with a root face different from the outer face.
The bijective construction presented here relies deeply on the theory of
\alpha-orientations introduced by Felsner, and in particular on the existence
of minimal and accessible orientations. Since most of the families of maps can
be characterized by such orientations, our generic bijective method is proved
to capture as special cases all previously known bijections involving
blossoming trees: for example Eulerian maps, m-Eulerian maps, non separable
maps and simple triangulations and quadrangulations of a k-gon. Moreover, it
also permits to obtain new bijective constructions for bipolar orientations and
d-angulations of girth d of a k-gon.
As for applications, each specialization of the construction translates into
enumerative by-products, either via a closed formula or via a recursive
computational scheme. Besides, for every family of maps described in the paper,
the construction can be implemented in linear time. It yields thus an effective
way to encode and generate planar maps.
In a recent work, Bernardi and Fusy introduced another unified bijective
scheme, we adopt here a different strategy which allows us to capture different
bijections. These two approaches should be seen as two complementary ways of
unifying bijections between planar maps and decorated trees.Comment: 45 pages, comments welcom
Tableau sequences, open diagrams, and Baxter families
Walks on Young's lattice of integer partitions encode many objects of
algebraic and combinatorial interest. Chen et al. established connections
between such walks and arc diagrams. We show that walks that start at
, end at a row shape, and only visit partitions of bounded height
are in bijection with a new type of arc diagram -- open diagrams. Remarkably
two subclasses of open diagrams are equinumerous with well known objects:
standard Young tableaux of bounded height, and Baxter permutations. We give an
explicit combinatorial bijection in the former case.Comment: 20 pages; Text overlap with arXiv:1411.6606. This is the full version
of that extended abstract. Conjectures from that work are proved in this wor
A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape
Tableau sequences of bounded height have been central to the analysis of
k-noncrossing set partitions and matchings. We show here that familes of
sequences that end with a row shape are particularly compelling and lead to
some interesting connections. First, we prove that hesitating tableaux of
height at most two ending with a row shape are counted by Baxter numbers. This
permits us to define three new Baxter classes which, remarkably, do not
obviously possess the antipodal symmetry of other known Baxter classes. We then
conjecture that oscillating tableau of height bounded by k ending in a row are
in bijection with Young tableaux of bounded height 2k. We prove this conjecture
for k at most eight by a generating function analysis. Many of our proofs are
analytic in nature, so there are intriguing combinatorial bijections to be
found.Comment: 10 pages, extended abstrac
New bijective links on planar maps via orientations
This article presents new bijections on planar maps. At first a bijection is
established between bipolar orientations on planar maps and specific
"transversal structures" on triangulations of the 4-gon with no separating
3-cycle, which are called irreducible triangulations. This bijection
specializes to a bijection between rooted non-separable maps and rooted
irreducible triangulations. This yields in turn a bijection between rooted
loopless maps and rooted triangulations, based on the observation that loopless
maps and triangulations are decomposed in a similar way into components that
are respectively non-separable maps and irreducible triangulations. This gives
another bijective proof (after Wormald's construction published in 1980) of the
fact that rooted loopless maps with edges are equinumerous to rooted
triangulations with inner vertices.Comment: Extended and revised journal version of a conference paper with the
title "New bijective links on planar maps", which appeared in the Proceedings
of FPSAC'08, 23-27 June 2008, Vi\~na del Ma
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