Enumerative Results Research Articles

On symmetric quadrangulations and triangulations

This article presents new enumerative results related to symmetric planar maps. In the first part a new way of enumerating rooted simple quadrangulations and rooted simple triangulations is presented, based on the description of two different quotient operations on symmetric simple quadrangulations and triangulations. In the second part, based on the results of Bouttier, Di Francesco and Guitter and on quotient and substitution operations, the series of three families of symmetric quadrangular and triangular dissections of polygons are computed, with control on the distance from the central vertex to the outer boundary.

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.

Actions and Identities on Set Partitions

A labeled set partition is a partition of a set of integers whose arcs are labeled by nonzero elements of an abelian group $\mathbb{A}$. Inspired by the action of the linear characters of the unitriangular group on its supercharacters, we define a group action of $\mathbb{A}^n$ on the set of $\mathbb{A}$-labeled partitions of an $(n+1)$-set. By investigating the orbit decomposition of various families of set partitions under this action, we derive new combinatorial proofs of Coker's identity for the Narayana polynomial and its type B analogue, and establish a number of other related identities. In return, we also prove some enumerative results concerning André and Neto's supercharacter theories of type B and D.

Combinatorial variations on Cantorʼs diagonal

We discuss counting problems linked to finite versions of Cantorʼs diagonal of infinite tableaux. We extend previous results of Brlek et al. (2004) [2] by refining an equivalence relation that reduces significantly the exhaustive generation. New enumerative results follow and allow to look at the sub-class of the so-called bi-Cantorian tableaux. We conclude with a correspondence between Cantorian-type tableaux and coloring of hypergraphs having a square number of vertices.

Blocking, efficiency and weighted optimality

Optimal blocking is explored for experiments, such as those incorporating one or more controls, where not all treatment comparisons are of equal interest. Weighted optimality functions are employed in gaining both analytic and enumerative results; a catalogue of smaller optimal designs is provided. It is shown how design selection based on functions of variances, and on functions of efficiency factors, are both subsumed by the weighted approach. Copyright 2011, Oxford University Press.

A bijection for triangulations, quadrangulations, pentagulations, etc.

A d-angulation is a planar map with faces of degree d. We present for each integer d ⩾ 3 a bijection between the class of d-angulations of girth d (i.e., with no cycle of length less than d) and a class of decorated plane trees. Each of the bijections is obtained by specializing a “master bijection” which extends an earlier construction of the first author. Our construction unifies known bijections by Fusy, Poulalhon and Schaeffer for triangulations ( d = 3 ) and by Schaeffer for quadrangulations ( d = 4 ). For d ⩾ 5 , both the bijections and the enumerative results are new. We also extend our bijections so as to enumerate p-gonal d-angulations ( d-angulations with a simple boundary of length p) of girth d. We thereby recover bijectively the results of Brown for simple p-gonal triangulations and simple 2 p-gonal quadrangulations and establish new results for d ⩾ 5 . A key ingredient in our proofs is a class of orientations characterizing d-angulations of girth d. Earlier results by Schnyder and by De Fraysseix and Ossona de Mendez showed that simple triangulations and simple quadrangulations are characterized by the existence of orientations having respectively indegree 3 and 2 at each inner vertex. We extend this characterization by showing that a d-angulation has girth d if and only if the graph obtained by duplicating each edge d − 2 times admits an orientation having indegree d at each inner vertex.

Homology of the boolean complex

We construct and analyze an explicit basis for the homology of the boolean complex of a finite simple graph. This provides a combinatorial description of the spheres in the wedge sum representing the homotopy type of the complex. We assign a set of derangements to any finite simple graph. For each derangement, we construct a corresponding element in the homology of the complex, and the collection of these elements forms a basis for the homology of the boolean complex. In this manner, the spheres in the wedge sum describing the homotopy type of the complex can be represented by a set of derangements. We give an explicit, closed-form description of the derangements that can be obtained from any finite simple graph, and compute this set for several families of graphs. In the cases of complete graphs and Ferrers graphs, these calculations give bijective proofs of previously obtained enumerative results.

Loop-free Gray code algorithm for the e-restricted growth functions

The subject of Gray codes algorithms for the set partitions of { 1 , 2 , … , n } had been covered in several works. The first Gray code for that set was introduced by Knuth (1975) [5], later, Ruskey presented a modified version of Knuthʼs algorithm with distance two, Ehrlich (1973) [3] introduced a loop-free algorithm for the set of partitions of { 1 , 2 , … , n } , Ruskey and Savage (1994) [9] generalized Ehrlichʼs results and give two Gray codes for the set of partitions of { 1 , 2 , … , n } , and recently, Mansour et al. (2008) [7] gave another Gray code and loop-free generating algorithm for that set by adopting plane tree techniques. In this paper, we introduce the set of e-restricted growth functions (a generalization of restricted growth functions) and extend the aforementioned results by giving a Gray code with distance one for this set; and as a particular case we obtain a new Gray code for set partitions in restricted growth function representation. Our Gray code satisfies some prefix properties and can be implemented by a loop-free generating algorithm using classical techniques; such algorithms can be used as a practical solution of some difficult problems. Finally, we give some enumerative results concerning the restricted growth functions of order d.

Two enumerative results on cycles of permutations

Answering a question of Bóna, it is shown that for n ≥ 2 the probability that 1 and 2 are in the same cycle of a product of two n -cycles on the set { 1 , 2 , … , n } is 1 / 2 if n is odd and 1 2 − 2 ( n − 1 ) ( n + 2 ) if n is even. Another result concerns the polynomial P λ ( q ) = ∑ w q κ ( ( 1 , 2 , … , n ) ⋅ w ) , where w ranges over all permutations in the symmetric group S n of cycle type λ , ( 1 , 2 , … , n ) denotes the n -cycle 1 → 2 → ⋯ → n → 1 , and κ ( v ) denotes the number of cycles of the permutation v . A formula is obtained for P λ ( q ) from which it is deduced that all zeros of P λ ( q ) have real part 0.

The structure of alternative tableaux

In this paper we study alternative tableaux introduced by Viennot [X. Viennot, Alternative tableaux, permutations and partially asymmetric exclusion process, talk in Cambridge, 2008]. These tableaux are in simple bijection with permutation tableaux, defined previously by Postnikov [A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764v1 [math.CO], 2006]. We exhibit a simple recursive structure for alternative tableaux, from which we can easily deduce a number of enumerative results. We also give bijections between these tableaux and certain classes of labeled trees. Finally, we exhibit a bijection with permutations, and relate it to some other bijections that already appeared in the literature.

Boltzmann Samplers, Pólya Theory, and Cycle Pointing

We introduce a general method to count unlabeled combinatorial structures and to efficiently generate them at random. The approach is based on pointing unlabeled structures in an “unbiased” way so that a structure of size $n$ gives rise to $n$ pointed structures. We extend Polya theory to the corresponding pointing operator and present a random sampling framework based on both the principles of Boltzmann sampling and Polya operators. All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results. Our method is illustrated in several examples: in each case, we provide enumerative results and efficient random samplers. The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families of planar maps.

Face rings of simplicial complexes with singularities

The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular, i.e., has the homology of a wedge of spheres of the expected dimension. This is derived from an enumerative result for local cohomology of face rings modulo generic linear forms, as compared with local cohomology of the face ring itself. The enumerative result is generalized to squarefree modules. A concept of Cohen–Macaulay in codimension c is defined and characterized for arbitrary finitely generated modules and coherent sheaves. For the face ring of an r-dimensional complex Δ, it is equivalent to nonsingularity of Δ in dimension r − c; for a coherent sheaf on projective space, this condition is shown to be equivalent to the same condition on any single generic hyperplane section. The characterization of nonsingularity in dimension m via finite local cohomology thus generalizes from face rings to arbitrary graded modules.

Rook placements in Young diagrams and permutation enumeration

Given two operators D ˆ and E ˆ subject to the relation D ˆ E ˆ − q E ˆ D ˆ = p , and a word w in D ˆ and E ˆ , the rewriting of w in normal form is combinatorially described by rook placements in Young diagrams. We give enumerative results about these rook placements, particularly in the case where p = ( 1 − q ) / q 2 . This case naturally arises in the context of the PASEP, a random process whose partition function and stationary distribution can be derived using two operators D and E subject to the relation D E − q E D = D + E (matrix Ansatz). Using the link obtained by Corteel and Williams between the PASEP, permutation tableaux and permutations, we prove a conjecture of Corteel and Rubey about permutation enumeration. This result gives the generating function for permutations of given size with respect to the number of ascents and occurrences of the pattern 13-2, this is also the moments of some q-Laguerre orthogonal polynomials.

Bijections for Baxter families and related objects

The Baxter number B n can be written as B n = ∑ k = 0 n Θ k , n − k − 1 with Θ k , ℓ = 2 ( k + 1 ) 2 ( k + 2 ) ( k + ℓ k ) ( k + ℓ + 1 k ) ( k + ℓ + 2 k ) . These numbers have first appeared in the enumeration of so-called Baxter permutations; B n is the number of Baxter permutations of size n, and Θ k , ℓ is the number of Baxter permutations with k descents and ℓ rises. With a series of bijections we identify several families of combinatorial objects counted by the numbers Θ k , ℓ . Apart from Baxter permutations, these include plane bipolar orientations with k + 2 vertices and ℓ + 2 faces, 2-orientations of planar quadrangulations with k + 2 white and ℓ + 2 black vertices, certain pairs of binary trees with k + 1 left and ℓ + 1 right leaves, and a family of triples of non-intersecting lattice paths. This last family allows us to determine the value of Θ k , ℓ as an application of the lemma of Lindström Gessel–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 plane bipolar orientations, also Schnyder woods of triangulations. Most of the enumerative results and some of the bijections are not new. Our contribution is mainly in the simplified and unifying presentation of this beautiful piece of combinatorics.

Cycle Lemma, Parking Functions and Related Multigraphs

For positive integers a and b, an $${(a, \overline{b})}$$-parking function of length n is a sequence (p 1, . . . , p n ) of nonnegative integers whose weakly increasing order q 1 ≤ . . . ≤ q n satisfies the condition q i < a + (i − 1)b. In this paper, we give a new proof of the enumeration formula for $${(a, \overline{b})}$$-parking functions by using of the cycle lemma for words, which leads to some enumerative results for the $${(a, \overline{b})}$$-parking functions with some restrictions such as symmetric property and periodic property. Based on a bijection between $${(a, \overline{b})}$$-parking functions and rooted forests, we enumerate combinatorially the $${(a, \overline{b})}$$-parking functions with identical initial terms and symmetric $${(a, \overline{b})}$$-parking functions with respect to the middle term. Moreover, we derive the critical group of a multigraph that is closely related to $${(a, \overline{b})}$$-parking functions.

On $k$-crossings and $k$-nestings of permutations

We introduce $k$-crossings and $k$-nestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint distribution. As a corollary, the number of $k$-noncrossing permutations is equal to the number of $k$-nonnesting permutations. We also provide some enumerative results for $k$-noncrossing permutations for some values of $k$. Nous introduisons les $k$-chevauchement d'arcs et les $k$-empilements d'arcs de permutations. Nous montrons que l'index de chevauchement et l'index de empilement ont une distribution conjointe symétrique pour les permutations de taille $n$. Comme corollaire, nous obtenons que le nombre de permutations n'ayant pas un $k$-chevauchement est égal au nombre de permutations n'ayant un $k$-empilement. Nous fournissons également quelques résultats énumératifs.

Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions

Given a finite irreducible Coxeter group W W , a positive integer d d , and types T 1 , T 2 , … , T d T_1,T_2,\dots ,T_d (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions c = σ 1 σ 2 ⋯ σ d c=\sigma _1\sigma _2\cdots \sigma _d of a Coxeter element c c of W W , such that σ i \sigma _i is a Coxeter element in a subgroup of type T i T_i in W W , i = 1 , 2 , … , d i=1,2,\dots ,d , and such that the factorisation is “minimal” in the sense that the sum of the ranks of the T i T_i ’s, i = 1 , 2 , … , d i=1,2,\dots ,d , equals the rank of W W . For the exceptional types, these decomposition numbers have been computed by the first author in [“Topics in Discrete Mathematics,” M. Klazar et al. (eds.), Springer–Verlag, Berlin, New York, 2006, pp. 93–126] and [Séminaire Lotharingien Combin. 54 (2006), Article B54l]. The type A n A_n decomposition numbers have been computed by Goulden and Jackson in [Europ. J. Combin. 13 (1992), 357–365], albeit using a somewhat different language. We explain how to extract the type B n B_n decomposition numbers from results of Bóna, Bousquet, Labelle and Leroux [Adv. Appl. Math. 24 (2000), 22–56] on map enumeration. Our formula for the type D n D_n decomposition numbers is new. These results are then used to determine, for a fixed positive integer l l and fixed integers r 1 ≤ r 2 ≤ ⋯ ≤ r l r_1\le r_2\le \dots \le r_l , the number of multi-chains π 1 ≤ π 2 ≤ ⋯ ≤ π l \pi _1\le \pi _2\le \dots \le \pi _l in Armstrong’s generalised non-crossing partitions poset, where the poset rank of π i \pi _i equals r i r_i and where the “block structure” of π 1 \pi _1 is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non-crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type D n D_n generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrong’s F = M F=M Conjecture in type D n D_n , thus completing a computational proof of the F = M F=M Conjecture for all types. It also allows one to address another conjecture of Armstrong on maximal intervals containing a random multi-chain in the generalised non-crossing partitions poset.

Some odd permutations

I recently stumbled on an identity (Theorem 2) that related two types of permutations. It looked like one of those serendipitous pieces of mathematics that led to a simple enumerative result; mostly when this happens it leads joyously to an enumerative proof that was ‘blindingly’ obvious. Except in this case, it wasn’t, I don’t think.

Hook Length Polynomials for Plane Forests of a Certain Type

The original motivation for the study of hook length polynomials was to find a combinatorial proof for a hook length formula for binary trees given by Postnikov, as well as a proof for a hook length polynomial formula conjectured by Lascoux. In this paper, we define the hook length polynomial for plane forests of a given degree sequence type and show that it can be factored into a product of linear forms. Some other enumerative results on forests are also given.

A Linear Programming Construction of Fully Optimal Bases in Graphs and Hyperplane Arrangements

The fully optimal basis of a bounded acyclic oriented matroid on a linearly ordered set has been defined and studied by the present authors in a series of papers, dealing with graphs, hyperplane arrangements, and oriented matroids (in order of increasing generality). This notion provides a bijection between bipolar orientations and uniactive internal spanning trees in a graph resp. bounded regions and uniactive internal bases in a hyperplane arrangement or an oriented matroid (in the sense of Tutte activities). This bijection is the basic case of a general activity preserving bijection between reorientations and subsets of an oriented matroid, called the active bijection, providing bijective versions of various classical enumerative results. Fully optimal bases can be considered as a strenghtening of optimal bases from linear programming, with a simple combinatorial definition. Our first construction used this purely combinatorial characterization, providing directly an algorithm to compute in fact the reverse bijection. A new definition uses a direct construction in terms of a linear programming. The fully optimal basis optimizes a sequence of nested faces with respect to a sequence of objective functions (whereas an optimal basis in the usual sense optimizes one vertex with respect to one objective function). This note presents this construction in terms of graphs and linear algebra.