This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures.
These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of context-free languages, 3. holonomicity (i.e., P-recursiveness) of numbers of labeled regular graphs, 4. frequent occurrence of the asymptotics cn^{-3/2}r^n and 5. ultimate modular periodicity of numbers of MSOL-definable structures.
This paper discusses the enumeration for the total number of all rooted spanning forests of the labeled complete tripartite graph. We enumerate the total number by a combinatorial decomposition.
Using the correspondence between a cycle up-down permutation and a pair of matchings, we give a combinatorial proof of the enumeration of alternating permutations according to the given peak set.
We introduce a one-person game that we call Padlock Solitaire which resembles the well-known clock solitaire card game. Analyzing variants of this game we obtain simple proofs of some classical results of combinatorics including ballot theorems and the enumeration of spanning trees in various graphs and hypergraphs.
In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled trees, and labeled plane trees.
In this survey we discuss some of the significant contributions of Ian Goulden and David Jackson in the areas of classical enumeration, symmetric functions, factorizations of permutations, and algebraic foundations of quantum field theory. Through their groundbreaking textbook, {\em Combinatorial Enumeration}, and their numerous research papers, both together and with their many students, they have had an influence in areas of bioinformatics, mathematical chemistry, algorithmic computer science, and theoretical physics. Here we review and set in context highlights of their 40 years of collabor…
A recent paper of Edman et al. has taken a combinatorial approach to measuring the anonymity of a threshold mix anonymous communications system. Their paper looks at ways of matching individual messages sent to individual messages received, irrespective of user, and determines a measure of the anonymity provided by the system. Here we extend this approach to include in the calculation information about how many messages were sent or received by a user and we define a new metric that can be computed exactly and efficiently using classical and elegant techniques from combinatorial enumeration.
We obtain a differential equation for the enumeration of the path length of general increasing trees. By using differential operators and their combinatorial interpretation we give a bijective proof of a version of Faà di Bruno formula, and model the generation of ballot and Dyck paths. We get formulas for its enumeration according with the height of their lattice points. Recursive formulas for the enumeration of enriched increasing trees and forests with respect to the height of their internal and external vertices are also obtained. Finally we present a generalized form of all those results …
We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d-simplex. Our generator has implications for several outstanding problems in polytope theory, including conjectures about the number of distinct polytopes, the edge expansion of polytopal graphs, and the d-step conjecture.
We show that, given a suitable combinatorial specification for a permutation class $\mathcal{C}$, one can obtain a specification for the juxtaposition (on either side) of $\mathcal{C}$ with Av(21) or Av(12), and that if the enumeration for $\mathcal{C}$ is given by a rational or algebraic generating function, so is the enumeration for the juxtaposition. Furthermore this process can be iterated, thereby providing an effective method to enumerate any 'skinny' $k\times 1$ grid class in which at most one cell is non-monotone, with a guarantee on the nature of the enumeration given the nature of th…
Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and Fukuda (2001) published the first database of oriented matroids including degenerate (i.e. non-uniform) ones and of higher ranks. In this paper, we investigate algorithmic ways to classify them in terms of realizability, although the underlying decision problem of realizability checking is NP-hard. As an application, we determine all possible combinatorial t…
It is well known that the set of $m$-Dyck paths with a fixed height and a fixed amount of valleys is counted by the Fuß-Narayana numbers. In this article, we consider the set of $m$-Dyck paths that start with at least $t$ north steps. We give exact formulas for the number of such paths with fixed height, fixed number of returns and (i) fixed number of valleys, (ii) fixed number of valleys with $x$-coordinate divisible by $m$ and (iii) fixed number of valleys with $x$-coordinate not divisible by $m$. The enumeration (ii) combinatorially realizes the $H$-triangle appearing in a recent article of…
A combinatorial formula is derived which expresses free cumulants in terms of classical comulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connected set partitions respectively. The proof relies on Moebius inversion on the partition lattice.
In enumerative combinatorics, it is often a goal to enumerate both labeled and unlabeled structures of a given type. The theory of combinatorial species is a novel toolset which provides a rigorous foundation for dealing with the distinction between labeled and unlabeled structures. The cycle index series of a species encodes the labeled and unlabeled enumerative data of that species. Moreover, by using species operations, we are able to solve for the cycle index series of one species in terms of other, known cycle indices of other species. Section 3 is an exposition of species theory and Sect…
This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and geometric definition, but are also related to some deep algebraic structures. For instance, a special type of maps called constellations provides a unifying framework for some enumeration problems concerning factorizations in the symmetric group. Standing on a position where many domains meet, maps can be studied using a large variety of methods, and their enumer…
The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns \cite{Raney}. We give a new combinatorial interpretation for all Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the $p$-Catalan numbers in terms of $p$-ary trees via the specialization $R_{p,1}(n) =_{p} c_n$. Our technique leads to several combinatorial identities involving the Raney numbers and ordered partitions. We then give addition…
We introduce a large family of combinatorial objects, called standard puzzles, defined by very simple rules. We focus on the standard puzzles for which the enumeration problems can be solved by explicit formulas or by classical numbers, such as binomial coefficients, Fibonacci numbers, tangent numbers, Catalan numbers, $\ldots$
Recently, Bényi and the second author introduced two combinatorial interpretations for symmetrized poly-Bernoulli polynomials. In the present study, we construct bijections between these combinatorial objects. We also define various combinatorial polynomials and prove that all of these polynomials coincide with symmetrized poly-Bernoulli polynomials.
Using the theory of combinatorial species, we compute the cycle index for bipartite graphs, which we use to count unlabeled bipartite graphs and bipartite blocks.
For enumerative problems, i.e. computable functions f from N to Z, we define the notion of an effective (or closed) formula. It is an algorithm computing f(n) in the number of steps that is polynomial in the combined size of the input n and the output f(n), both written in binary notation. We discuss many examples of enumerative problems for which such closed formulas are, or are not, known. These problems include (i) linear recurrence sequences and holonomic sequences, (ii) integer partitions, (iii) pattern-avoiding permutations, (iv) triangle-free graphs and (v) regular graphs. In part I we …
The enumeration of combinatorial classes of the complex polynomial vector fields in C presented in [Dia13] is extended here to a closed form enumeration of combinatorial classes for degree d polynomial vector fields up to rotations of 2(d-1)st roots of unity. The main tool in the proof of this result is based on a general method of enumeration developed by V.A.Liskovets [Lis98].
In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in the tree can get multiple labels. For all tree classes we show characterizations of suitable generating functions for the tree enumeration sequence via differential equations. Furthermore, for several combinatorial classes of multilabelled increasing tree families we present explicit enumeration results. We also present multilabelled increasing tree familie…
We study set partitions with $r$ distinguished elements and block sizes found in an arbitrary index set $S$. The enumeration of these $(S,r)$-partitions leads to the introduction of $(S,r)$-Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the $r$-Stirling numbers. We also introduce the associated $(S,r)$-Bell and $(S,r)$-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For $S$ with some extra structure, we show that the inverse of the $(S,r)$-Stirling matrix encodes th…
In this paper we investigate enumeration of some classes of $n$-character strings and binary necklaces. Recall that binary necklaces are necklaces in two colors with length $n$. We prove three results (Theorems 1, 1' and 2) concerning the numbers of three classes of $j$-character strings (closely related to some classes of binary necklaces or Lyndon words). Using these results, we deduce Moreau's necklace-counting function for binary aperiodic necklaces of length $k$ \cite{mo} (Theorem 3), and we prove the binary case of MacMahon's formula from 1892 \cite{ma} (also called Witt's formula) for t…
We describe an algorithm to enumerate polytopes. This algorithm is then implemented to give a complete classification of combinatorial spheres of dimension 3 with 9 vertices and decide polytopality of those spheres. In particular, we completely enumerate all combinatorial types of 4-dimensional polytopes with 9 vertices. It is shown that all of those combinatorial types are rational: They can be realized with rational coordinates. We find 316014 combinatorial spheres on 9 vertices. Of those, 274148 can be realized as the boundary complex of a four-dimensional polytope and the remaining 41866 a…
We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in terms of elementary symmetric functions. Analysis of some of the combinatorial implications of this expansion leads to three bijections involving acyclic orientations.
Springer numbers are an analog of Euler numbers for the group of signed permutations. Arnol'd showed that they count some objects called snakes, that generalize alternating permutations. Hoffman established a link between Springer numbers, snakes, and some polynomials related with the successive derivatives of trigonometric functions. The goal of this article is to give further combinatorial properties of derivative polynomials, in terms of snakes and other objects: cycle-alternating permutations, weighted Dyck or Motzkin paths, increasing trees and forests. We obtain the generating functions,…
We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the initial counting sequence and for asymptotic analysis. The key tool is the scalar product for symmetric functions and that this operation preserves D-finiteness.
We derive combinatorial identities, involving the Bernoulli and Euler numbers, for the numbers of standard Young tableaux of certain skew shapes. This generalizes the classical formulas of D. Andre on the number of up-down permutations. The analysis uses a transfer operator approach extending the method of Elkies, combined with an identity expressing the volume of a certain polytope in terms of a Schur function.
Dyck paths where peaks are only allowed on level 1 and on even-indexed levels, were introduced by Retakh and analysed by Zeilberger, with assistance from Ekhad. We add some combinatorial comments to the enumeration, which involves Motzkin numbers, in particular, about the average height of such objects.
In this paper, we enumerate lattice paths with certain constraints and apply the corresponding results to develop formulas for calculating the dimensions of submodules of a class of modules for planar upper triangular rook monoids. In particular, we show that the famous Catalan numbers appear as the dimensions of some special modules; we also obtain some combinatorial identities
Point-determining graphs are graphs in which no two vertices have the same neighborhoods, co-point-determining graphs are those whose complements are point-determining, and bi-point-determining graphs are those both point-determining and co-point-determining. Bicolored point-determining graphs are point-determining graphs whose vertices are properly colored with white and black. We use the combinatorial theory of species to enumerate these graphs as well as the connected cases.
Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds: cycle covers, closed walks, spanning trees. We show that if the weights of the edges of the graph are within a constant factor, fixed in advance, of each other then the bulk of the partition function is concentrated on the subsets of a particularly simple structure: cycle covers with few cycles, walks that visit every vertex only few times, and spanning trees wi…
We study the enumeration problem for different kind of tree parking functions introduced recently, called tree parking functions, tree parking distributions, prime tree parking functions, and prime tree parking distributions, for rooted labelled trees of important combinatorial tree families including labelled ordered, unordered and binary trees. Using combinatorial decompositions of the underlying structures yields, after solving the resulting equations, implicit characterizations of suitable generating functions of the total number of such tree parking functions for trees of size $n$ and $n$…
The Gaussian polynomial in variable $q$ is defined as the $q$-analog of the binomial coefficient. In addition to remarkable implications of these polynomials to abstract algebra, matrix theory and quantum computing, there is also a combinatorial interpretation through weighted lattice paths. This interpretation is equivalent to weighted board tilings, which can be used to establish Gaussian polynomial identities. In particular, we prove duals of such identities and evaluate related sums.
We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from which the coefficients can be extracted. As a byproduct, we also enumerate labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps.
We give combinatorial proofs of the formulas for the number of multichains in the $k$-divisible noncrossing partitions of classical types with certain conditions on the rank and the block size due to Krattenthaler and M{ü}ller. We also prove Armstrong's conjecture on the zeta polynomial of the poset of $k$-divisible noncrossing partitions of type $A$ invariant under a $180^\circ$ rotation in the cyclic representation.
Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. The first result of this paper is a $(q,t)$-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. Our proof uses commutation relations for $(q,t)$-vertex operators acting on Macdonald polynomials as given by Garsia, Haiman and Tesla. The second result of this paper is an explicit combinatorial interpreation of the $(q,t)$-Macdonald weight in term…
A connected graph is 2K2-free if it does not contain a pair of independent edges as an induced subgraph. In this paper, we present the structural characterization of minimal vertex separator and show that there are polynomial number of minimal vertex separators in 2K2-free graphs. Further, using the enumeration we show that finding minimum connected vertex separator in 2K2-free graphs is polynomial time solvable. We highlight that finding minimum connected vertex separator is NP-complete in Chordality 5 graphs, which is a super graph class of 2K2-free graphs. Other study includes, enumeration …
We study the class of graphs known as k-trees through the lens of Joyal's theory of combinatorial species (and an equivariant extension known as '$Γ$-species' which incorporates data about 'structural' group actions). This culminates in a system of recursive functional equations giving the generating function for unlabeled k-trees which allows for fast, efficient computation of their numbers. Enumerations up to k = 10 and n = 30 (for a k-tree with (n+k-1) vertices) are included in tables, and Sage code for the general computation is included in an appendix.
We describe an algorithm for the enumeration of (candidates of) vertex-transitive combinatorial $d$-manifolds. With an implementation of our algorithm, we determine, up to combinatorial equivalence, all combinatorial manifolds with a vertex-transitive automorphism group on $n\leq 13$ vertices. With the exception of actions of groups of small order, the enumeration is extended to 14 and 15 vertices.
This paper is the second in a series of planned papers which provide first bijective proofs of alternating sign matrix results. Based on the main result from the first paper, we construct a bijective proof of the enumeration formula for alternating sign matrices and of the fact that alternating sign matrices are equinumerous with descending plane partitions. We are also able to refine these bijections by including the position of the unique $1$ in the top row of the matrix. Our constructions rely on signed sets and related notions. The starting point for these constructions were known ``comput…
A ($κ$,$τ$)-regular set is a vertex subset S inducing a $κ$-regular subgraph such that every vertex out of S has $τ$ neighbors in S. This article is an expository overview of the main results obtained for graphs with ($κ$,$τ$)-regular sets. The graphs with classical combinatorial structures, like perfect matchings, Hamilton cycles, efficient dominating sets, etc, are characterized by ($κ$,$τ$)-regular sets whose determination is equivalent to the determination of those classical combinatorial structures. The characterization of graphs with these combinatorial structures are presented. The dete…
We provide a new strategy to compute the exponential growth constant of enumeration sequences counting walks in lattice path models restricted to the quarter plane. The bounds arise by comparison with half-planes models. In many cases the bounds are provably tight, and provide a combinatorial interpretation of recent formulas of Fayolle and Raschel (2012) and Bostan, Raschel and Salvy (2013). We discuss how to generalize to higher dimensions.
Following the work of Cano and Diaz, we consider a continuous analog of lattice path enumeration. This allows us to define a continuous version of any discrete object that counts certain types of lattice paths. We define continuous versions of binomials and multinomials, and describe some identities and partial differential equations they satisfy. Finally, we illustrate a general process to recover discrete combinatorial quantities from their continuous analogs.
We introduce a natural weighted enumeration of lattice points in a polytope, and give a Brion-type formula for the corresponding generating function. The weighting has combinatorial significance, and its generating function may be viewed as a generalization of the Rogers-Szegő polynomials. It also arises from the geometry of the toric arc scheme associated to the normal fan of the polytope. We show that the asymptotic behavior of the coefficients at $q = 1$ is Gaussian.
Hochschild lattices are specific intervals in the dexter meet-semilattices recently introduced by Chapoton. A natural geometric realization of these lattices leads to some cell complexes introduced by Saneblidze, called the Hochschild polytopes. We obtain several geometrical properties of the Hochschild lattices, namely we give cubic realizations, establish that these lattices are EL-shellable, and show that they are constructible by interval doubling. We also prove several combinatorial properties as the enumeration of their $k$-chains and compute their degree polynomials.
Signed Counting on Restricted Partitions and Combinatorial Proofs of Three Identities
College of Mathematics and Statistics & Center for Discrete Mathematics, Chongqing University（重庆大学数学与统计学院与离散数学中心）；College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）
We enumerate the edges in the Hasse diagram of several lattices arising in the combinatorial context of lattice paths. Specifically, we will consider the case of Dyck, Grand Dyck, Motzkin, Grand Motzkin, Schröder and Grand Schröder lattices. Finally, we give a general formula for the number of edges in an arbitrary Young lattice (which can be interpreted in a natural way as a lattice of paths).
