Class Of Combinatorial Objects Research Articles

Combinatorial generation via permutation languages. II. Lattice congruences

This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called quotientopes, a family of polytopes recently introduced by Pilaud and Santos [Bull. Lond. Math. Soc., 51:406–420, 2019], which generalize permutahedra, associahedra, hypercubes and several other polytopes. We prove that all of these graphs have a Hamilton path, which can be computed by a simple greedy algorithm. This is an application of our framework for exhaustively generating various classes of combinatorial objects by encoding them as permutations. We also characterize which of these graphs are vertex-transitive or regular via their arc diagrams, give corresponding precise and asymptotic counting results, and we determine their minimum and maximum degrees.

The Rank Function of a Positroid and Non-Crossing Partitions

A positroid is a special case of a realizable matroid, that arose from the study of totally nonnegative part of the Grassmannian by Postnikov. Postnikov demonstrated that positroids are in bijection with certain interesting classes of combinatorial objects, such as Grassmann necklaces and decorated permutations. The bases of a positroid can be described directly in terms of the Grassmann necklace and decorated permutation. In this paper, we show that the rank of an arbitrary set in a positroid can be computed directly from the associated decorated permutation using non-crossing partitions.

Combinatorial identities for tenth order mock theta functions

In this paper, the open problem posed by Sareen and Rana (Proc. Indian Acad. Sci. (Math. Sci.) 126 (2016) 549–556) is addressed. Here, we interpret two tenth order mock theta functions combinatorially in terms of lattice paths. Then we extend enumeration of one of these with Bender–Knuth matrices; the other by using Frobenius partitions. The combinatorial interpretation of one of these mock theta functions in terms of Frobenius partitions gives an answer to the open problem. Finally, we establish bijections between different classes of combinatorial objects which lead us to one 4-way and one 3-way combinatorial identity.

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).

Non-planar on-shell diagrams

We initiate a systematic study of non-planar on-shell diagrams in N=4 SYM and develop powerful technology for doing so. We introduce canonical variables generalizing face variables, which make the dlog form of the on-shell form explicit. We make significant progress towards a general classification of arbitrary on-shell diagrams by means of two classes of combinatorial objects: generalized matching and matroid polytopes. We propose a boundary measurement that connects general on-shell diagrams to the Grassmannian. Our proposal exhibits two important and non-trivial properties: positivity in the planar case and it matches the combinatorial description of the diagrams in terms of generalized matroid polytopes. Interestingly, non-planar diagrams exhibit novel phenomena, such as the emergence of constraints on Plucker coordinates beyond Plucker relations when deleting edges, which are neatly captured by the generalized matching and matroid polytopes. This behavior is tied to the existence of a new type of poles in the on-shell form at which combinations of Plucker coordinates vanish. Finally, we introduce a prescription, applicable beyond the MHV case, for writing the on-shell form as a function of minors directly from the graph.

Ordered partitions and drawings of rooted plane trees

We study the bounded regions in a generic slice of the hyperplane arrangement in Rn consisting of the hyperplanes defined by xi and xi+xj. The bounded regions are in bijection with several classes of combinatorial objects, including the ordered partitions of [n] all of whose left-to-right minima occur at odd locations and the drawings of rooted plane trees with n+1 vertices. These are sequences of rooted plane trees such that each tree in a sequence can be obtained from the next one by removing a leaf.

Elementary proofs of Radu and Sellers’ results for broken 2-diamond partitions

In 2007, Andrews and Paule introduced a new class of combinatorial objects called broken k-diamonds. Let Δk(n) be the number of broken k-diamonds partitions of n. By using MacMahon’s partition analysis, they found that the generating function of Δk(n) is an infinite product, which attracted many mathematicians to study its congruence properties. Recently, Radu and Sellers found infinitely many Ramanujan-like congruences modulo 3 for Δ2(n). In fact, they obtained 3-dissections of the generating function for Δ2(n) modulo 3 by using the theory of modular forms. In this brief note, we aim to present elementary proofs of Radu and Sellers’ results by using several theta function identities due to Ramanujan.

Factorizations of Cycles and Multi-Noded Rooted Trees

In this paper, we study factorizations of cycles. The main result is that under certain condition, the number of ways to factor a d-cycle into a product of cycles of prescribed lengths is d r-2. To prove our result, we first define a new class of combinatorial objects, multi-noded rooted trees, which generalize rooted trees. We find the cardinality of this new class which with proper parameters is exactly d r-2. The main part of this paper is the proof that there is a bijection from factorizations of a d-cycle to multi-noded rooted trees via factorization graphs. This implies the desired formula. The factorization problem we consider has its origin in geometry, and is related to the study of a special family of Hurwitz numbers: pure-cycle Hurwitz numbers. Via the standard translation of Hurwitz numbers into group theory, our main result is equivalent to the following: when the genus is 0 and one of the ramification indices is d, the degree of the covers, the pure-cycle Hurwitz number is d r-3, where r is the number of branch points.

Formula omitted]-tilings and triangulations of the strip

SL2-tilings were introduced by Assem, Reutenauer, and Smith in connection with frieses and their applications to cluster algebras. An SL2-tiling is a bi-infinite matrix of positive integers such that each adjacent 2×2-submatrix has determinant 1.In this paper we define the class of SL2-tilings with enough ones. It contains the previously known tilings as well as some which are new, and we show that it is in bijection with a certain class of combinatorial objects, namely “good” triangulations of the strip.

Combinatorial trees arising in the study of interval exchange transformations

In this paper we study a new class of combinatorial objects that we call trees of relations equipped with an operation that we call induction. These trees were first introduced in Ferenczi and Zamboni (2010) [3] in the context of interval exchange transformations but they may be studied independently from a purely combinatorial point of view. They possess a variety of interesting combinatorial properties and have already been linked to a number of different areas including ergodic theory and number theory—see Ferenczi and Zamboni (2010, in press) [3,4]. In a recent sequel to this paper, Marsh and Schroll have established interesting connections to the theory of cluster algebras and polygonal triangulations: Marsh and Schroll (2010) [5]. For each tree of relations G, we let Γ(G) denote the smallest set of trees of relations containing G and invariant under induction. The induction mapping allows us to endow Γ(G) with the structure of a connected directed graph, which we call the graph of graphs. We investigate the structure of Γ(G) and define a circular order based on the tree structure which turns out to be a complete invariant for the induction mapping. This gives a complete characterization of Γ(G) which allows us to compute its cardinality in terms of Catalan numbers. We show that the circular order also defines an abstract secondary structure similar to one occurring in genetics in the study of RNA.

De Bruijn cycles for neural decoding

Stimulus counterbalance is critical for studies of neural habituation, bias, anticipation, and (more generally) the effect of stimulus history and context. We introduce de Bruijn cycles, a class of combinatorial objects, as the ideal source of pseudo-random stimulus sequences with arbitrary levels of counterbalance. Neuro-vascular imaging studies (such as BOLD fMRI) have an additional requirement imposed by the filtering and noise properties of the method: only some temporal frequencies of neural modulation are detectable. Extant methods of generating counterbalanced stimulus sequences yield neural modulations that are weakly (or not at all) detected by BOLD fMRI. We solve this limitation using a novel “path-guided” approach for the generation of de Bruijn cycles. The algorithm encodes a hypothesized neural modulation of specific temporal frequency within the seemingly random order of events. By positioning the modulation between the signal and noise bands of the neuro-vascular imaging method, the resulting sequence markedly improves detection power. These sequences may be used to study stimulus context and history effects in a manner not previously possible.

$n!$ matchings, $n!$ posets

We show that there are matchings on points without so-called left (neighbor) nestings. We also define a set of naturally labeled -free posets and show that there are such posets on elements. Our work was inspired by Bousquet-Melou, Claesson, Dukes and Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884-909]. They gave bijections between four classes of combinatorial objects: matchings with no neighbor nestings (due to Stoimenow), unlabeled -free posets, permutations avoiding a specific pattern, and so-called ascent sequences. We believe that certain statistics on our matchings and posets could generalize the work of Bousquet-Melou et al., and we make a conjecture to that effect. We also identify natural subsets of matchings and posets that are equinumerous to the class of unlabeled -free posets. We give bijections that show the equivalence of (neighbor) restrictions on nesting arcs with (neighbor) restrictions on crossing arcs. These bijections are thought to be of independent interest. One of the bijections factors through certain upper-triangular integer matrices that have recently been studied by Dukes and Parviainen [Electron. J. Combin. 17 (2010) #R53].

Triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ and Tropical Oriented Matroids

Develin and Sturmfels showed that regular triangulations of $\Delta_{n-1} \times \Delta_{d-1}$ can be thought of as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of $\Delta_{n-1} \times \Delta_{d-1}$. In this paper, we show that any triangulation of $\Delta_{n-1} \times \Delta_{d-1}$ encodes a tropical oriented matroid. We also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes. Develin et Sturmfels ont montré que les triangulations de $\Delta_{n-1} \times \Delta_{d-1}$ peuvent être considérées comme des polytopes tropicaux. Les matroïdes orientés tropicaux ont été définis par Ardila et Develin, et ils ont été conjecturés être en bijection avec les subdivisions de $\Delta_{n-1} \times \Delta_{d-1}$. Dans cet article, nous montrons que toute triangulation de $\Delta_{n-1} \times \Delta_{d-1}$ encode un matroïde orienté tropical. De plus, nous proposons une nouvelle classe d'objets combinatoires qui peuvent décrire toutes les subdivisions d'une plus grande classe de polytopes.

Formula omitted]-free posets, ascent sequences and pattern avoiding permutations

We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled ( 2 + 2 ) -free posets and a certain class of involutions (or chord diagrams), already appeared in the literature, but were apparently not known to be equinumerous. We present a direct bijection between them. The third class is a family of permutations defined in terms of a new type of pattern. An attractive property of these patterns is that, like classical patterns, they are closed under the action of the symmetry group of the square. The fourth class is formed by certain integer sequences, called ascent sequences, which have a simple recursive structure and are shown to encode ( 2 + 2 ) -free posets and permutations. Our bijections preserve numerous statistics. We determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for the class of chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern 3 1 ¯ 52 4 ¯ and use this to enumerate those permutations, thereby settling a conjecture of Pudwell.

$n!$ matchings, $n!$ posets (extended abstract)

We show that there are $n!$ matchings on $2n$ points without, so called, left (neighbor) nestings. We also define a set of naturally labelled $(2+2)$-free posets, and show that there are $n!$ such posets on $n$ elements. Our work was inspired by Bousquet-Mélou, Claesson, Dukes and Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884―909]. They gave bijections between four classes of combinatorial objects: matchings with no neighbor nestings (due to Stoimenow), unlabelled $(2+2)$-free posets, permutations avoiding a specific pattern, and so called ascent sequences. We believe that certain statistics on our matchings and posets could generalize the work of Bousquet-Mélou et al. and we make a conjecture to that effect. We also identify natural subsets of matchings and posets that are equinumerous to the class of unlabeled $(2+2)$-free posets. We give bijections that show the equivalence of (neighbor) restrictions on nesting arcs with (neighbor) restrictions on crossing arcs. These bijections are thought to be of independent interest. One of the bijections maps via certain upper-triangular integer matrices that have recently been studied by Dukes and Parviainen [Electron. J. Combin. 17 (2010) #R53]. Nous montrons qu'il y a $n!$ couplages sur $2n$ points sans emboîtement (de voisins) à gauche. Nous définissons aussi un ensemble d'EPO (ensembles partiellement ordonnés) sans motif $(2+2)$ naturellement étiquetés, et montrons qu'il y a $n!$ tels EPO sur $n$ éléments. Notre travail a été inspiré par Bousquet-Mélou, Claesson, Dukes et Kitaev [J. Combin. Theory Ser. A. 117 (2010) 884―909]. Ces auteurs donnent des bijections entre quatre classes d'objets combinatoires: couplages sans emboîtement de voisins (dû à Stoimenow), EPO sans motif $(2+2)$ non étiquetés, permutations évitant un certain motif, et des objets appelés suites à montées. Nous pensons que certaines statistiques sur nos couplages et nos EPO pourraient généraliser le travail de Bousquet-Mélou et al. et nous proposons une conjecture à ce sujet. Nous identifions aussi des sous-ensembles naturels de couplages et d'EPO qui sont énumérés par la même séquence que la classe des EPO sans motif $(2+2)$ non étiquetés. Nous donnons des bijections qui démontrent l'équivalence entre les restrictions sur les emboîtements (d'arcs voisins) et les restrictions sur les croisements (d'arcs voisins). Nous pensons que ces bijections présentent un intérêt propre. L'une de ces bijections passe par certaines matrices triangulaires supérieures à coefficients entiers qui ont été récemment étudiées par Dukes et Parviainen [Electron. J. Combin. 17 (2010) #R53].

Unlabeled $(2+2)$-free posets, ascent sequences and pattern avoiding permutations

We present statistic-preserving bijections between four classes of combinatorial objects. Two of them, the class of unlabeled $(\textrm{2+2})$-free posets and a certain class of chord diagrams (or involutions), already appeared in the literature, but were apparently not known to be equinumerous. The third one is a new class of pattern avoiding permutations, and the fourth one consists of certain integer sequences called $\textit{ascent sequences}$. We also determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern $3\bar{1}52\bar{4}$, and enumerate those permutations, thus settling a conjecture of Pudwell. Nous présentons des bijections, transportant de nombreuses statistiques, entre quatre classes d'objets. Deux d'entre elles, la classe des EPO (ensembles partiellement ordonnés) sans motif $(\textrm{2+2})$ et une certaine classe d'involutions, sont déjà apparues dans la littérature. La troisième est une classe de permutations à motifs exclus, et la quatrième une classe de suites que nous appelons $\textit{suites à montées}$. Nous déterminons ensuite la série génératrice de ces classes, retrouvant ainsi un résultat prouvé par Zagier pour les involutions sus-mentionnées. La série obtenue n'est pas D-finie. Apparemment, le fait qu'elle compte aussi les EPO sans motif $(\textrm{2+2})$ est nouveau. Finalement, nous caractérisons les suites à montées qui correspondent aux permutations évitant le motif barré $3\bar{1}52\bar{4}$ et énumérons ces permutations, ce qui démontre une conjecture de Pudwell.

Recursive Constructions of Detecting Matrices for Multiuser Coding: A Unifying Approach

Detecting matrices are a class of combinatorial objects originated from the coin weighing problem of Soderberg and Shapiro in the early 1960s. In this paper, various known recursive construction techniques for binary, bipolar, and ternary detecting matrices are reexamined in a unifying framework. New, general recursive constructions of detecting matrices, which include previous recursive constructions as special cases, are derived. Such matrices find applications in multiuser coding since they are equivalent to a certain class of uniquely decodable multiuser codes for the binary adder channel. Interestingly, it is found that among the three kinds of detecting matrices, ternary detecting matrices are of fundamental significance from the combinatorial theoretic, as well as from the multiuser coding application, point of view.

Generating effective symmetry-breaking predicates for search problems

Consider the problem of testing for existence of an n-node graph G satisfying some condition P, expressed as a Boolean constraint among the n × n Boolean entries of the adjacency matrix M. This problem reduces to satisfiability of P ( M ) . If P is preserved by isomorphism, P ( M ) is satisfiable iff P ( M ) ∧ SB ( M ) is satisfiable, where SB ( M ) is a symmetry-breaking predicate—a predicate satisfied by at least one matrix M in each isomorphism class. P ( M ) ∧ SB ( M ) is more constrained than P ( M ) , so it is solved faster by backtracking than P ( M ) —especially if SB ( M ) rules out most matrices in each isomorphism class. This method, proposed by Crawford et al., applies not just to graphs but to testing existence of a combinatorial object satisfying any property that respects isomorphism, as long as the property can be compactly specified as a Boolean constraint on the object's binary representation. We present methods for generating symmetry-breaking predicates for several classes of combinatorial objects: acyclic digraphs, permutations, functions, and arbitrary-arity relations (direct products). We define a uniform optimality measure for symmetry-breaking predicates, and evaluate our constraints according to this measure. Results indicate that these constraints are either optimal or near-optimal for their respective classes of objects.

Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm

Synthesis of kinematic chains can be viewed as the enumeration of a certain class of graphs. Very efficient algorithms, using group theoretic techniques, exist for exhaustive isomorph-free generation of certain classes of combinatorial objects, which either eliminate or restrict the explicit isomorphism detection. An algorithm belonging to one particular class called McKay-type, in combination with an efficient degeneracy testing algorithm, is used for the synthesis of planar mechanisms. The existing results are reexamined and the discrepancies are reconciled.

Locally 2‐arc transitive graphs, homogeneous factorizations, and partial linear spaces

AbstractWe establish natural bijections between three different classes of combinatorial objects; namely certain families of locally 2‐arc transitive graphs, partial linear spaces, and homogeneous factorizations of arc‐transitive graphs. Moreover, the bijections intertwine the actions of the relevant automorphism groups. Thus constructions in any of these areas provide examples for the others. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 139–148, 2006