Non-separable Planar Maps Research Articles

From generalized Tamari intervals to non-separable planar maps

Let v be a grid path made of north and east steps. The lattice TAM(v), based on all grid paths weakly above the grid path v sharing the same endpoints as v, was introduced by Pre ́ville-Ratelle and Viennot (2014) and corresponds to the usual Tamari lattice in the case v = (NE)n. They showed that TAM(v) is isomorphic to the dual of TAM(←−v ), where ←−v is the reverse of v with N and E exchanged. Our main contribution is a bijection from intervals in TAM(v) to non-separable planar maps. It follows that the number of intervals in TAM(v) over all v of length n is 2(3n+3)! (n+2)!(2n+3)! . This formula was first obtained by Tutte(1963) for non-separable planar maps.

A trinity of duality: Non-separable planar maps, β(1,0)-trees and synchronized intervals

The dual of a map is a fundamental construction on combinatorial maps, but many other combinatorial objects also possess their notion of duality. For instance, the Tamari lattice is isomorphic to its order dual, which induces an involution on the set of so-called “synchronized intervals” introduced by Préville-Ratelle and the present author. Another example is the class of β(1,0)-trees, which has a mysterious involution h proposed by Claesson, Kitaev and Steingrímsson (2009). These two classes of combinatorial objects are all in bijection with the class of non-separable planar maps, which is closed under map duality. In this article, we show that we can identify the notions of duality in these three classes using previously known natural bijections, which leads to a bijective proof of a result from Kitaev and de Mier (2013).

The enumeration of generalized Tamari intervals

Let v be a grid path made of north and east steps. The lattice Tam(v), based on all grid paths weakly above v and sharing the same endpoints as v, was introduced by Préville-Ratelle and Viennot (2016) and corresponds to the usual Tamari lattice in the case v=(NE)n. Our main contribution is that the enumeration of intervals in Tam(v), over all v of length n, is given by 2(3n+3)!(n+2)!(2n+3)!. This formula was first obtained by Tutte (1963) for the enumeration of non-separable planar maps. Moreover, we give an explicit bijection from these intervals in Tam(v) to non-separable planar maps.

Gray coding cubic planar maps

The idea of (combinatorial) Gray codes is to list objects in question in such a way that two successive objects differ in some pre-specified small way. In this paper, we utilize β-description trees to cyclicly Gray code three classes of cubic planar maps, namely, bicubic planar maps, 3-connected cubic planar maps, and cubic non-separable planar maps.

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

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.

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, β(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 study how certain structures in planar maps transfer to trees and permutations via the bijections. More precisely, we show that the number of 2-faces in a map equals the number of nodes in the corresponding β(1,0)-tree that are single children with maximum label; 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 Steingrímsson in 2009, to show that 2-face-free maps correspond to permutations avoiding certain mesh patterns. Finally, we give asymptotics for some of our enumerative results.

Decompositions and statistics for [formula omitted]-trees and nonseparable permutations

The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice through a stack (where the letters on the stack have to be in increasing order) was conjectured by West, and later proved by Zeilberger. Goulden and West found a bijection from such permutations to nonseparable planar maps, and later, Jacquard and Schaeffer presented a bijection from these planar maps to certain labeled plane trees, called β ( 1 , 0 ) -tree. Using generating trees, Dulucq, Gire and West showed that nonseparable planar maps are equinumerous with permutations avoiding the (classical) pattern 2413 and the barred pattern 41 3 ¯ 52 ; they called these permutations nonseparable. We give a new bijection between β ( 1 , 0 ) -tree and permutations avoiding the dashed patterns 3-1-4-2 and 2-41-3. These permutations can be seen to be exactly the reverse of nonseparable permutations. Our bijection is built using decompositions of the permutations and the trees, and it translates seven statistics on the trees into statistics on the permutations. Among the statistics involved are ascents, left-to-right minima and right-to-left maxima for the permutations, and leaves and the rightmost and leftmost paths for the trees. In connection with this we give a nontrivial involution on the β ( 1 , 0 ) -tree, which specializes to an involution on unlabeled rooted plane trees, where it yields interesting results. Lastly, we conjecture the existence of a bijection between nonseparable permutations and two-stack sortable permutations preserving at least four permutation statistics.

Baxter permutations and plane bipolar orientations

Abstract We present a simple bijection between Baxter permutations of size n and plane bipolar orientations with n edges. This bijection translates several classical parameters of permutations into natural parameters of orientations, and has remarkable symmetry properties. By specialising it to Baxter permutations avoiding the pattern 3142, we obtain a bijection with non-separable planar maps, which had been described only in an implicit recursive manner so far (up to simple symmetries). A further specialization yields a bijection between permutations avoiding 3142 and 2413 and series-parallel maps.

A Bijective Census of Nonseparable Planar Maps

Bijections are obtained between nonseparable planar maps and two different kinds of trees: description trees and skew ternary trees. A combinatorial relation between the latter and ternary trees allows bijective enumeration and random generation of nonseparable planar maps. The involved bijections take account of the usual combinatorial parameters and give a bijective proof of formulae established by Brown and Tutte. These results, combined with a bijection due to Goulden and West, give a purely combinatorial enumeration of two-stack-sortable permutations.

A combinatorial proof of J. West's conjecture

We give what we believe to be the first combinatorial proof of J. West's conjecture that is to say we show that two-stack sortable permutations are in correspondence with rooted nonseparable planar maps. The generating tree of a set of permutations with forbidden subsequences and its characterization by a rewriting system are used to obtain this correspondence. Other results on the number of two-stack sortable permutations with given classical parameters are presented.

Raney Paths and a Combinatorial Relationship between Rooted Nonseparable Planar Maps and Two-Stack-Sortable Permutations

An encoding of the set of two-stack-sortable permutations (TSS) in terms of lattice paths and ordered lists of strings is obtained. These lattice paths are called Raney paths. The encoding yields combinatorial decompositions for two complementary subsets of TSS, which are the analogues of previously known decompositions for the set of nonseparable rooted planar maps (NS). This provides a combinatorial relationship between TSS and NS, and, hence, a bijection is determined between these sets that is different, simpler, and more refined than the previously known bijection.

Permutations with forbidden subsequences and nonseparable planar maps

The goal of the present work is to connect combinatorially a family of maps to a family of permutations with forbidden subsequences. We obtain a generating tree of nonseparable planar rooted maps and show that this tree is the generating tree of a family of permutations. The distribution of these permutations is then obtained. Finally, the different steps leading to the combinatorial proof of West's conjecture are listed.

On chromatic and dichromatic sum equations

This paper provides firstly an exposition on the theory of chromatic sums for rooted planar triangulations founded by W.T. Tutte. Then, the generalization of the theory for rooted nonseparable planar maps is described. Further, a functional equation of the dichromatic sum functions for rooted nonseparable planar maps is also established by using the same decomposition lemmas as those in the chromatic case. However, it seems that the equations obtained here can not be directly derived from the dichromatic sum equation for rooted general planar maps which was discovered by W.T. Tutte as well.

On the Enumeration of Rooted Non-Separable Planar Maps

It has been shown elsewhere (1, 4) that the number of rooted non-separable planar maps with n edges isIn the present paper we improve upon this result by finding the number fi,j of rooted non-separable planar maps with i + 1 vertices and j + 1 faces. We use the definitions of (1).Among the non-separable planar maps only the loop-map and the link-map have i = 0 or j = 0. We therefore confine our attention to the case in which i and j are both positive.