Direct Bijection Research Articles

K-Factorizations of the full cycle and generalized Mahonian statistics on k-forests

We develop direct bijections between the set Fnk of minimal factorizations of the long cycle (01⋯kn) into (k+1)-cycle factors and the set Rnk of rooted labelled forests on vertices {1,…,n} with edges coloured with {0,1,…,k−1} that map natural statistics on the former to generalized Mahonian statistics on the latter. In particular, we examine the generalized major index on forests Rnk and show that it has a simple and natural interpretation in the context of factorizations. Our results extend those in [8], which treated the case k=1 through a different approach, and provide a bijective proof of the equidistribution observed by Yan [16] between displacement of k-parking functions and generalized inversions of k-forests.

The Canonical Bijection between Pipe Dreams and Bumpless Pipe Dreams

Abstract We present a direct bijection between reduced pipe dreams and reduced bumpless pipe dreams (BPDs) by interpreting reduced compatible sequences on BPDs and show that this is the unique bijection preserving bijective realizations of Monk’s formula, establishing its canonical nature.

Bijections between planar maps and planar linear normal λ-terms with connectivity condition

The enumeration of linear λ-terms has attracted quite some attention recently, partly due to their link to combinatorial maps. Zeilberger and Giorgetti (2015) [19] gave a recursive bijection between planar linear normal λ-terms and planar maps, which, when restricted to 2-connected λ-terms (i.e., without closed sub-terms), leads to bridgeless planar maps. Inspired by this restriction, Zeilberger and Reed (2019) [20] conjectured that 3-connected planar linear normal λ-terms have the same counting formula as bipartite planar maps. In this article, we settle this conjecture by giving a direct bijection between these two families. Furthermore, using a similar approach, we give a direct bijection between planar linear normal λ-terms and planar maps, whose restriction to 2-connected λ-terms leads to loopless planar maps. This bijection seems different from that of Zeilberger and Giorgetti, even after taking the map dual. We also explore enumerative consequences of our bijections.

A bijection between Tamari intervals and extended fighting fish

We introduce extended fighting fish as branching surfaces that can also be seen as walks in the quarter plane defined by simple rewriting rules. The main result we present in the article is a direct bijection between extended fighting fish and intervals of the Tamari lattice that exchanges multiple natural statistics. The model includes the recently introduced fighting fish of (Duchi et al., 2017) that were shown to be equinumerated with synchronized Tamari intervals.Using the dual surface/walk points of view on extended fighting fish, we show that the area statistic on these fish corresponds to the distance statistic (or maximal length of a chain) in Tamari intervals. We also show that the average area of a uniform random extended fighting fish of size n, and hence the average distance over the set of Tamari intervals of size n, is of order n5/4, in accordance with earlier results for the subclass of ordinary fighting fish.

On a conjecture of Lin and Kim concerning a refinement of Schröder numbers

In this paper, we compute the distribution of the first letter statistic on nine avoidance classes of permutations corresponding to two pairs of patterns of length four. In particular, we show that the distribution is the same for each class and is given by the entries of a new Schroder number triangle. This answers in the affirmative a recent conjecture of Lin and Kim. We employ a variety of techniques to prove our results, including generating trees, direct bijections and the kernel method. For the latter, we make use of in a creative way what we are trying to show in three cases to aid in solving a system of functional equations satisfied by the associated generating functions.

The Canonical Complex of the Weak Order

We define and study the canonical complex of a finite semidistributive lattice L. It is the simplicial complex on the join or meet irreducible elements of L which encodes each interval of L by recording the canonical join representation of its bottom element and the canonical meet representation of its top element. This complex behaves properly with respect to lattice quotients of L, in the sense that the canonical complex of a quotient of L is the subcomplex of the canonical complex of L induced by the join or meet irreducibles of L uncontracted in the quotient. We then describe combinatorially the canonical complex of the weak order on permutations in terms of semi-crossing arc bidiagrams, formed by the superimposition of two non-crossing arc diagrams of N. Reading. We provide explicit direct bijections between the semi-crossing arc bidiagrams and the weak order interval posets of G. Châtel, V. Pilaud and V. Pons. Finally, we provide an algorithm to describe the Kreweras maps in any lattice quotient of the weak order in terms of semi-crossing arc bidiagrams.

Avoiding a pair of patterns in multisets and compositions

In this paper, we study the Wilf-type equivalence relations among multiset permutations. We identify all multiset equivalences among pairs of patterns consisting of a pattern of length three and another pattern of length at most four. To establish our results, we make use of a variety of techniques, including Ferrers-equivalence arguments, sorting by minimal/maximal letters, analysis of active sites and direct bijections. In several cases, our arguments may be extended to prove multiset equivalences for infinite families of pattern pairs. Our results apply equally well to the Wilf-type classification of compositions, and as a consequence, we obtain a complete description of the Wilf-equivalence classes for pairs of patterns of type (3,3) and (3,4) on compositions, with the possible exception of two classes of type (3,4).

Bijective link between Chapoton’s new intervals and bipartite planar maps

In 2006, Chapoton defined a class of Tamari intervals called ”new intervals” in his enumeration of Tamari intervals, and he found that these new intervals are equi-enumerated with bipartite planar maps. We present here a direct bijection between these two classes of objects using a new object called ”degree tree”. Our bijection also gives an intuitive proof of an unpublished equi-distribution result of some statistics on new intervals given by Chapoton and Fusy.

On q-Series and Split Lattice Paths

In this paper a natural question which arise to study the graphical aspect of split $$(n+t)$$ -color partitions, is answered by introducing a new class of lattice paths, called split lattice paths. A direct bijection between split $$(n+t)$$ -color partitions and split lattice paths is proved. This new combinatorial object is applied to give new combinatorial interpretations of two basic functions of Gordon-McIntosh. Some generalized q-series are also discussed. We further explore these paths by providing combinatorial interpretations of some Rogers-Ramanujan type identities which reveal their rich structure and great potential for further research.

The number of corner polyhedra graphs

Corner polyhedra were introduced by Eppstein and Mumford (2014) as the set of simply connected 3D polyhedra such that all vertices have non negative integer coordinates, edges are parallel to the coordinate axes and all vertices but one can be seen from infinity in the direction (1, 1, 1). These authors gave a remarkable characterization of the set of corner polyhedra graphs, that is graphs that can be skeleton of a corner polyhedron: as planar maps, they are the duals of some particular bipartite triangulations, which we call hereafter corner triangulations.In this paper we count corner polyhedral graphs by determining the generating function of the corner triangulations with respect to the number of vertices: we obtain an explicit rational expression for it in terms of the Catalan gen- erating function. We first show that this result can be derived using Tutte's classical compositional approach. Then, in order to explain the occurrence of the Catalan series we give a direct algebraic decomposition of corner triangu- lations: in particular we exhibit a family of almond triangulations that admit a recursive decomposition structurally equivalent to the decomposition of binary trees. Finally we sketch a direct bijection between binary trees and almond triangulations. Our combinatorial analysis yields a simpler alternative to the algorithm of Eppstein and Mumford for endowing a corner polyhedral graph with the cycle cover structure needed to realize it as a polyhedral graph.

The Abelian sandpile model on Ferrers graphs — A classification of recurrent configurations

We classify all recurrent configurations of the Abelian sandpile model (ASM) on Ferrers graphs. The classification is in terms of decorations of EW-tableaux, which undecorated are in bijection with the minimal recurrent configurations. We introduce decorated permutations, extending to decorated EW-tableaux a bijection between such tableaux and permutations, giving a direct bijection between the decorated permutations and all recurrent configurations of the ASM. We also describe a bijection between the decorated permutations and the intransitive trees of Postnikov, the breadth-first search of which corresponds to a canonical toppling of the corresponding configurations.

Planar triangulations, bridgeless planar maps and Tamari intervals

We present a direct bijection between planar 3-connected triangulations and bridgeless planar maps, which were first enumerated by Tutte (1962) and Walsh and Lehman (1975) respectively. Previously known bijections by Wormald (1980) and Fusy (2010) are all defined recursively. Our direct bijection passes by a new class of combinatorial objects called “sticky trees”. We also present bijections between sticky trees, intervals in the Tamari lattices and closed flows on forests. With our bijections, we recover several known enumerative results about these objects. We thus show that sticky trees can serve as a nexus of bijective links among all these equi-enumerated objects.

Nine classes of permutations enumerated by binomial transform of Fine’s sequence

The problem of avoidance of a single permutation pattern or of a pair of patterns of length four has been well studied. Less is known concerning the avoidance of three 4-letter patterns. In this paper, we determine up to symmetry all triples of 4-letter patterns such that the number of members of Sn avoiding any one of them is given by the binomial transform of Fine’s sequence (see A033321 in OEIS). We make use of both algebraic and combinatorial proofs in order to establish our results. In a couple of cases, we introduce certain auxiliary statistics on Sn which give rise to a system of functional equations that can be solved using the kernel method. In another case, a direct bijection is defined between members of the avoidance class in question and the set of skew Dyck paths.

A FAMILY OF THE ZECKENDORF THEOREM RELATED IDENTITIES

In this paper we present a family of identities for recursive sequences arising from a second order recurrence relation, that gives instances of Zeckendorf representation. We prove these results using a special case of an universal property of the recursive sequences. In particular cases we also establish a direct bijection. Besides, we prove further equalities that provide a representation of the sum of (r + 1)-st and (r − 1)-st Fibonacci number as the sum of powers of the golden ratio. Similarly, we show a class of natural numbers represented as the sum of powers of the silver ratio. AMS Subject Classification: 11B39, 11B37

Symmetry and Log-Concavity Results for Statistics on Fibonacci Tableaux

In this paper, we study the properties of the inversion statistic and the Fibonacci major index, Fibmaj, as defined on standard Fibonacci tableaux. We prove that these two statistics are symmetric and log-concave over all standard Fibonacci tableaux of a given shape μ and provide two combinatorial proofs of the symmetry result, one a direct bijection on the set of tableaux and the other utilizing 0, 1-fillings of a staircase shape. We conjecture that the inversion and Fibmaj statistics are log-concave over all standard Fibonacci tableaux of a given size n. In addition, we show a well-known bijection between standard Fibonacci tableaux of size n and involutions in Sn which takes the Fibmaj statistic to a new statistic called the submajor index on involutions.

Triangulations of Cayley and Tutte polytopes

Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun’s conjecture, which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to two one-variable deformations of Cayley polytopes (which we call t-Cayley and t-Gayley polytopes), and to the most general two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph.Our approach is based on an explicit triangulation of the Cayley and Tutte polytopes. We prove that simplices in the triangulations correspond to labeled trees. The heart of the proof is a direct bijection based on the neighbors-first search graph traversal algorithm.

A combinatorial way of counting unicellular maps and constellations

Our work is devoted to the bijective enumeration of the set of factorizations of a permutation into m factors with a given number of cycles. Previously, this major problem in combinatorics and its various specializations were considered mainly from the character theoretic or algebraic geometry point of view. Let us specially mention here the works of Harer and Zagier or Kontsevich. In 1988, Jackson reported a very general formula solving this problem. However, to the author’s own admission this result left little room for combinatorial interpretation and no bijective proof of it was known yet. In 2001, Lass found a combinatorial proof of the celebrated special case of Jackson’s formula known as the Harer–Zagier formula. This work was followed by Goulden and Nica, who presented in 2004 another combinatorial proof involving a direct bijection. In the past two years, we have introduced new sets of objects called partitioned maps and partitioned cacti, the enumeration of which allowed us to construct bijective proofs for more general cases of Jackson’s formula.

A bijection between permutations and a subclass of TSSCPPs

We define a subclass of totally symmetric self-complementary plane partitions (TSSCPPs) which we show is in direct bijection with permutation matrices. This bijection maps the inversion number of the permutation, the position of the 1 in the last column, and the position of the 1 in the last row to natural statistics on these TSSCPPs. We also discuss the possible extension of this approach to finding a bijection between alternating sign matrices and all TSSCPPs. Finally, we remark on a new poset structure on TSSCPPs arising from this perspective which is a distributive lattice when restricted to permutation TSSCPPs.

Long Cycle Factorizations: Bijective Computation in the General Case

This paper is devoted to the computation of the number of ordered factorizations of a long cycle in the symmetric group where the number of factors is arbitrary and the cycle structure of the factors is given. Jackson (1988) derived the first closed form expression for the generating series of these numbers using the theory of the irreducible characters of the symmetric group. Thanks to a direct bijection we compute a similar formula and provide the first purely combinatorial evaluation of these generating series. Cet article est dédié au calcul du nombre de factorisations d’un long cycle du groupe symétrique pour lesquels le nombre de facteurs est arbitraire et la structure des cycles des facteurs est donnée. Jackson (1988) a dérivé la première expression compacte pour les séries génératrices de ces nombres en utilisant la théorie des caractères irréductibles du groupe symétrique. Grâce à une bijection directe nous démontrons une formule similaire et donnons ainsi la première évaluation purement combinatoire de ces séries génératrices.

Cayley and Tutte polytopes

Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun's conjecture, which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to a two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph. Our approach is based on an explicit triangulation of the Cayley and Tutte polytope. We prove that simplices in the triangulations correspond to labeled trees and forests. The heart of the proof is a direct bijection based on the neighbors-first search graph traversal algorithm. Les polytopes de Cayley ont été définis récemment comme des ensembles convexes de compositions de Cayley introduits par Cayley en 1857. Dans ce papier, nous résolvons la conjecture de Braun. Cette dernière exprime le volume du polytopes de Cayley en termes du nombre de graphes connexes. Nous étendons ce résultat à des déformations de polytopes de Cayley à deux variables, à savoir les polytopes de Tutte. Le volume de ces derniers est donnè par une évaluation du polynôme de Tutte du graphe complet. Notre approche est basée sur une triangulation explicite des polytopes de Cayley et Tutte. Nous démontrons que les simplexes de ces triangulations correspondent à des arbres marqués. La pierre angulaire de notre démonstration est une bijection directe basées sur l'algorithme de la recherche du premier voisin sur le graphe.