Baxter permutations

Chung et al. (1978) have proved that the number of Baxter permutations on [n] is

Viennot (1981) has then given a combinatorial proof of this formula, showing this sum corresponds to the distribution of these permutations according to their number of rises.

Cori et al. (1986), by making a correspondence between two families of planar maps, have shown that the number of alternating Baxter permutations on [2n+δ] is c_n+δc_n where c_n = (2n)!/(n + 1)!n! is the nth Catalan number.

In this paper, we establish a new one-to-one correspondence between Baxter permutations and three non-intersecting paths, which unifies Viennot (1981) and Cori et al. (1986). Moreover, we obtain more precise results for the enumeration of (alternating or not) Baxter permutations according to various parameters. So, we give a combinatorial interpretation of Mallows's formula (1979).     

Baxter permutations rise again

Baxter permutations, so named by Boyce, were introduced by Baxter in his study of the fixed points of continuous functions which commute under composition. Recently Chung, Graham, Hoggatt, and Kleiman obtained a sum formula for the number of Baxter permutations of 2n ? 1 objects, but admit to having no interpretation of the individual terms of this sum. We show that in fact the kth term of this sum counts the number of (reduced) Baxter permutations that have exactly k ? 1 rises.     

Bijective counting of plane bipolar orientations and Schnyder woods

A bijection Φ is presented between plane bipolar orientations with prescribed numbers of vertices and faces, and non-intersecting triples of upright lattice paths with prescribed extremities. This yields a combinatorial proof of the following formula due to Baxter for the number Θ_ij of plane bipolar orientations with i non-polar vertices and j inner faces:

In addition, it is shown that Φ specializes into the bijection of Bernardi and Bonichon between Schnyder woods and non-crossing pairs of Dyck words.This is the extended and revised journal version of a conference paper with the title “Bijective counting of plane bipolar orientations”, which appeared in Electr. Notes in Discr. Math. pp. 283–287 (Proceedings of Eurocomb’07, 11–15 September 2007, Sevilla).     

Restricted 123-avoiding Baxter permutations and the Padovan numbers

Baxter studied a particular class of permutations by considering fixed points of the composite of commuting functions. This class is called Baxter permutations. In this paper we investigate the number of 123-avoiding Baxter permutations of length n that also avoid (or contain a prescribed number of occurrences of) another certain pattern of length k. In several interesting cases the generating function depends only on k and is expressed via the generating function for the Padovan numbers.     

Rectilinear planar layouts and bipolar orientations of planar graphs

We propose a linear-time algorithm for generating a planar layout of a planar graph. Each vertex is represented by a horizontal line segment and each edge by a vertical line segment. All endpoints of the segments have integer coordinates. The total space occupied by the layout is at mostn by at most 2n–4. Our algorithm, a variant of one by Otten and van Wijk, generally produces a more compact layout than theirs and allows the dual of the graph to be laid out in an interlocking way. The algorithm is based on the concept of abipolar orientation. We discuss relationships among the bipolar orientations of a planar graph.Research partly supported by the Agence de L'Informatique du Ministere de L'Industrie, France, under contract No. 83-285.     

Simple permutations and pattern restricted permutations

A simple permutation is one that does not map any non-trivial interval onto an interval. It is shown that, if the number of simple permutations in a pattern restricted class of permutations is finite, the class has an algebraic generating function and is defined by a finite set of restrictions. Some partial results on classes with an infinite number of simple permutations are given. Examples of results obtainable by the same techniques are given; in particular it is shown that every pattern restricted class properly contained in the 132-avoiding permutations has a rational generating function.     

Cell rotation graphs of strongly connected orientations of plane graphs with an application

The cell rotation graph D(G) on the strongly connected orientations of a 2-edge-connected plane graph G is defined. It is shown that D(G) is a directed forest and every component is an in-tree with one root; if T is a component of D(G), the reversions of all orientations in T induce a component of D(G), denoted by T⁻, thus (T,T⁻) is called a pair of in-trees of D(G); G is Eulerian if and only if D(G) has an odd number of components (all Eulerian orientations of G induce the same component of D(G)); the width and height of T are equal to that of T⁻, respectively. Further it is shown that the pair of directed tree structures on the perfect matchings of a plane elementary bipartite graph G coincide with a pair of in-trees of D(G). Accordingly, such a pair of in-trees on the perfect matchings of any plane bipartite graph have the same width and height.     

Unfair permutations

We study unfair permutations, which are generated by letting $n$ players draw numbers and assuming that player $i$ draws $i$ times from the unit interval and records her largest value. This model is natural in the context of partitions: the score of the $i$th player corresponds to the multiplicity of the summand $i$ in a random partition, with the roles of minimum and maximum interchanged. We study the distribution of several parameters, namely the position of player $i$, the number of inversions, and the number of ascents. To perform some of the heavy computations, we use the computer algebra package Sigma.     

Baxter algebras and Hopf algebras

By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.

     

Counting permutations

We give a direct count of the number of permutations of n objects for which (a) all the cycles have lengths divisible by a fixed integer d, and (b) none of the cycles has length divisible by d.     

Parity-alternating permutations and successions

The study of parity-alternating permutations of {1, 2, … n} is extended to permutations containing a prescribed number of parity successions — adjacent pairs of elements of the same parity. Several enumeration formulae are computed for permutations containing a given number of parity successions, in conjunction with further parity and length restrictions. The objects are classified using direct construction and elementary combinatorial techniques. Analogous results are derived for circular permutations.     

Braids and partial permutations

We study the inverse braid monoid IBn introduced by Easdown and Lavers in 2004. We completely describe the factorizable structure of IBn and use this to give a new proof of the Easdown-Lavers presentation; we also derive several new presentations, each of which gives rise to a new presentation of the symmetric inverse monoid. We then define and study the pure inverse braid monoid IPn which is related to IBn in the same way that the pure braid group is related to the braid group.     

Arc permutations

Arc permutations and unimodal permutations were introduced in the study of triangulations and characters. This paper studies combinatorial properties and structures on these permutations. First, both sets are characterized by pattern avoidance. It is also shown that arc permutations carry a natural affine Weyl group action, and that the number of geodesics between a distinguished pair of antipodes in the associated Schreier graph, and the number of maximal chains in the weak order on unimodal permutations, are both equal to twice the number of standard Young tableaux of shifted staircase shape. Finally, a bijection from non-unimodal arc permutations to Young tableaux of certain shapes, which preserves the descent set, is described and applied to deduce a conjectured character formula of Regev.