In 1997, Bousquet-Mélou and Eriksson stated a broad generalization of Euler's distinct-odd partition theorem, namely the $(k,l)$-Euler theorem. Their identity involved the $(k,l)$-lecture-hall partitions, which, unlike usual difference conditions of partitions in Rogers-Ramanujan type identities, satisfy some ratio constraints. In a 2008 paper, in response to a question suggested by Richard Stanley, Savage and Yee provided a simple bijection for the $l$-lecture-hall partitions (the case $k=l$), whose specialization in $l=2$ corresponds to Sylvester's bijection. Subsequently, as an open question, a generalization of their bijection was suggested for the case $k,l\geq 2$. In the spirit of Savage and Yee's work, we provide and prove in this paper slight variations of the suggested bijection, not only for the case $k,l\geq 2$ but also for the cases $(k,1)$ and $(1,k)$ with $k\geq 4$. Furthermore, we show that our bijections equal the recursive bijections given by Bousquet-Mélou and Eriksson in their recursive proof of the $(k,l)$-lecture hall and finally provide the analogous recursive bijection for the $(k,l)$-Euler theorem.
Consider generalizations of the boundary singularities B_n of the functions on the real line to the case where the boundary consists of a finite number of l points. These singularities B_n^l could also arise in higher dimensional case, when the boundary is an immersed hypersurface. We obtain a particular recurrent equation on the numbers of connected components of very nice M-morsification spaces of the multiboundary singularities B_n^l. This helps us to express the numbers K_n^l (for l=2,3,4...) by Bernoulli-Euler numbers. We also find the corresponding generating functions.
A topological version of a longstanding conjecture of H. Hopf, originally proposed by W. Thurston, states that the sign of the Euler characteristic of a closed aspherical manifold of dimension $d=2m$ depends only on the parity of $m$. Gromov defined several hyperbolization functors which produce an aspherical manifold from a given simplicial or cubical manifold. We investigate the combinatorics of several of these hyperbolizations and verify the Euler Characteristic Sign Conjecture for each of them. In addition, we explore further combinatorial properties of these hyperbolizations as they rela…
For all positive integers $k,l,n$, the Little Glaisher theorem states that the number of partitions of $n$ into parts not divisible by $k$ and occurring less than $l$ times is equal to the number of partitions of $n$ into parts not divisible by $l$ and occurring less than $k$ times. While this refinement of Glaisher theorem is easy to establish by computation of the generating function, there is still no one-to-one canonical correspondence explaining it. Our paper brings an answer to this open problem through an arithmetical approach. Furthermore, in the case $l=2$, we discuss the possibility …
In this paper, we present a variant of the Balog-Szemerédi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product phenomenon.
At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer the combinatorial question on some particular relation of Euler numbers proved by Heneghan-Petersen, Power series for up-down min-max permutations, College Math. Journal, Vol. 45, No. 2 (2014), 83-91.
The usage of elementary submodels is a simple but powerful method to prove theorems, or to simplify proofs in infinite combinatorics. First we introduce all the necessary concepts of logic, then we prove classical theorems using elementary submodels. We also present a new proof of Nash-Williams's theorem on cycle-decomposition of graphs, and finally we improve a decomposition theorem of Laviolette concerning bond-faithful decompositions of graphs.
In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan's "Lost" Notebook, we obtain weighted forms of Euler's theorem on partitions with odd parts and distinct parts. This work is inspired by the insight of Andrews on the connection between Ramanujan's identities and Euler's theorem. Our combinatorial formulations of Ramanujan's identities rely on the notion of rooted partitions. Iterated Dyson's map and Sylvester's bijection are the main ingredients in the weighted forms of Euler's theorem.
In this paper we extend the block combinatorics partition theorems of Hindman and Milliken in the setting of the recursive system of the block Schreier families (B^xi) consisting of families defined for every countable ordinal xi. Results contain (a) a block partition Ramsey theorem for every countable ordinal xi (Hindman's theorem corresponding to xi=1, and Milliken's theorem to xi a finite ordinal), (b) a countable ordinal form of the block Nash-Williams partition theorem, and (c) a countable ordinal block partition theorem for sets closed in the infinite block analogue of Ellentuck's topolo…
We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding…
We obtain a unification of two refinements of Euler's partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt's insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.
We prove that the number of even parts and the number of times that parts are repeated have the same distribution over integer partitions with a fixed perimeter. This refines Straub's analog of Euler's Odd-Distinct partition theorem. We generalize the two concerned statistics to these of the part-difference less than $d$ and the parts not congruent to $1$ modulo $d+1$ and prove a distribution inequality, that has a similar flavor as Alder's ex-conjecture, over partitions with a prescribed perimeter. Both of our results are proved analytically and combinatorially.
This paper considers three separate matrices associated to graphs and (each dimension of) cell complexes. It relates all the coefficients of their respective characteristic polynomials to the geometric and combinatorial enumeration of three kinds of subobjects. The matrices are: the mesh matrix for integral d-cycles of Trent, the mesh matrix for integral d-boundaries, and the Kirchhoff matrix, i.e., the combinatorial Laplacian, for integral (d-1)-chains. Relations to Reidemeister-Franz torsion are elucidated and relations to the foundational work of R. Lyons and G. Kalai.
We investigate the combinatorial structure of subspaces of the exterior algebra of a finite-dimensional real vector space, working in parallel with the extremal combinatorics of hypergraphs. Using initial monomials, projections of the underlying vector space onto subspaces, and the interior product, we find analogues of local and global LYM inequalities, the Erdős-Ko-Rado theorem, and the Ahlswede-Khachatrian bound for $t$-intersecting hypergraphs. Using these tools, we prove a new extension of the Two Families Theorem of Bollobás, giving a weighted bound for subspace configurations satisfying…
We examine and clarify in detail the contributions of Yoshisuke Matsunaga (1694?--1744) to the computation of Bell numbers in the eighteenth century (in the Edo period), providing modern perspectives to some unknown materials that are by far the earliest in the history of Bell numbers. Later clarification and developments by Yoriyuki Arima (1714--1783), and several new results such as the asymptotic distributions (notably the corresponding local limit theorems) of a few closely related sequences are also given.
In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a $1 \times \infty$ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a $q$-series identity. Using this method, they recover quite a few classical $q$-series identities, but Euler's Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating fu…
This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formula for a class of `continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of `continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of t…
The main topic of this paper is various "hyperbolic" generalizations of the Edmonds-Rado theorem on the rank of intersection of two matroids. We prove several results in this direction and pose a few questions. We also give generalizations of the Obreschkoff theorem and recent results of J. Borcea and B. Shapiro.
We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give an explicit sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials. This gives a lot of corollaries including the existence of free filtrations. The proof is based on the result about the form of minimal generators of a logarithmic derivation module of a multiarrangement which satisfies the $b_2$-equality.
We discuss two theorems in analytic number theory and combinatory analysis that have seen increased use in recent years. A corollary to a Tauberian theorem of Ingham allows one to quickly prove asymptotic formulas for arithmetic sequences, so long as the corresponding generating function exhibits exponential growth of a certain form near its radius of convergence. Two common methods for proving the required analytic behavior are modular transformations and Euler-Maclaurin summation. However, these results are sometimes stated without certain technical conditions that are necessary for the comp…
In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Terao's celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally …
We study the complexity of computational problems arising from existence theorems in extremal combinatorics. For some of these problems, a solution is guaranteed to exist based on an iterated application of the Pigeonhole Principle. This results in the definition of a new complexity class within TFNP, which we call PLC (for "polynomial long choice"). PLC includes all of PPP, as well as numerous previously unclassified total problems, including search problems related to Ramsey's theorem, the Sunflower theorem, the Erdős-Ko-Rado lemma, and König's lemma. Whether the first two of these four prob…
A useful heuristic in the understanding of large random combinatorial structures is the Arratia-Tavare principle, which describes an approximation to the joint distribution of component-sizes using independent random variables. The principle outlines conditions under which the total variation distance between the true joint distribution and the approximation should be small, and was successfully exploited by Pittel in the cases of integer partitions and set partitions. We provide sufficient conditions for this principle to be true in a general context, valid for certain discrete probability di…
This paper generalizes the basic notions of additive and multiplicative combinatorics to the setting of group actions: if $G$ is a group acting on a set $X$, and we have subsets $A\subseteq G$ and $Y\subseteq X$ such that the set of pairs $g\cdot y$ with $g\in A,y\in Y$ is not much larger than $Y$, what structure must $A$ and $Y$ have? Briefly, what is the structure of sets with small "image set"? In this setting, we develop analogs of Ruzsa's triangle inequality, covering theorems, multiplicative energy, and the Balog-Szemerédi-Gowers theorem. Approximate stabilizers, which we call symmetry s…
An important problem from invariant theory is to describe the subspace of a tensor power of a representation invariant under the action of the group. According to Weyl's classic, the first main (later: 'fundamental') theorem of invariant theory states that all invariants are expressible in terms of a finite number among them, whereas a second main theorem determines the relations between those basic invariants. Here we present a transparent, combinatorial proof of a second fundamental theorem for the defining representation of the symplectic group Sp(2n). Our formulation is completely explicit…
We prove Euler's theorem of number theory developing an argument based on quandles. A quandle is an algebraic structure whose axioms mimic the three Reidemeister moves of knot theory.
We adapt methods coming from additive combinatorics in groups to the study of linear span in associative unital algebras. In particular, we establish for these algebras analogues of Diderrich-Kneser's and Hamidoune's theorems on sumsets and Tao's theorem on sets of small doubling. In passing we classify the finite-dimensional algebras over infinite fields with finitely many subalgebras. These algebras play a crucial role in our linear version of Diderrich-Kneser's theorem. We also explain how the original theorems for groups we linearize can be easily deduced from our results applied to group …
Many classical theorems in combinatorics establish the emergence of substructures within sufficiently large collections of objects. Well-known examples are Ramsey's theorem on monochromatic subgraphs and the Erdős-Rado sunflower lemma. Implicit versions of the corresponding total search problems are known to be PWPP-hard; here "implici" means that the collection is represented by a poly-sized circuit inducing an exponentially large number of objects. We show that several other well-known theorems from extremal combinatorics - including Erdős-Ko-Rado, Sperner, and Cayley's formula - give rise t…
We generalise Euler's partition theorem involving odd parts and different parts for all moduli and provide new companions to Rogers-Ramanujan- Andrews-Gordon identities related to this theorem.
In this paper, we give a conjecture, which generalises Euler's partition theorem involving odd parts and different parts for all moduli. We prove this conjecture for two family partitions. We give $q$-difference equations for the related generating function if the moduli is three. We provide new companions to Rogers-Ramanujan-Andrews-Gordon identities under this conjecture.
Motivated by the resemblance of a multivariate series identity and a finite analogue of Euler's pentagonal number theorem, we study multiple extensions of the latter formula. In a different direction we derive a common extension of this multivariate series identity and two formulas of Lucas. Finally we give a combinatorial proof of Lucas' formulas.
Regarding Euler's odd-strict theorem, which is the most basic partition identity, A refinement was done by Sylvester, and it was generalized by Bessenrodt to the r-regular and r-class regular cases. In this paper, we focus on the periodicity of the exponent seen on the r-regular side in Bessenrodt's generalization, and further generalize from that point of view. At the end of this paper, we also introduce a partition identity when extended without regard to the periodicity of the exponent.
The Boros-Moll polynomials arise in the evaluation of a quartic integral. The original double summation formula does not imply the fact that the coefficients of these polynomials are positive. Boros and Moll proved the positivity by using Ramanujan's Master Theorem to reduce the double sum to a single sum. Based on the structure of reluctant functions introduced by Mullin and Rota along with an extension of Foata's bijection between Meixner endofunctions and bi-colored permutations, we find a combinatorial proof of the positivity. In fact, from our combinatorial argument one sees that it is es…
Additive combinatorics is built around the famous theorem by Szemerédi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemerédi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a genera…
Ultrafilters are a tool, originating in mathematical logic and general topology, that has steadily found more and more uses in multiple areas of mathematics, such as combinatorics, dynamics, and algebra, among others. The purpose of this article is to introduce ultrafilters in a friendly manner and present some applications to the branch of combinatorics known as Ramsey theory, culminating with a new ultrafilter-based proof of van der Waerden's theorem.
The study of intersection problems in Extremal Combinatorics dates back perhaps to 1938, when Paul Erdős, Chao Ko and Richard Rado proved the (first) `Erdős-Ko-Rado theorem' on the maximum possible size of an intersecting family of $k$-element subsets of a finite set. Since then, a plethora of results of a similar flavour have been proved, for a range of different mathematical structures, using a wide variety of different methods. Structures studied in this context have included families of vector subspaces, families of graphs, subsets of finite groups with given group actions, and of course u…
For each positive integer $n$, we construct a bijection between the odd partitions and the distinct partitions of $n$ which extends Bressoud's bijection between the odd-and-distinct partitions of $n$ and the splitting partitions of $n$. We compare our bijection with the classical bijections of Glaisher and Sylvester, and also with a recent bijection due to Chen, Gao, Ji and Li.
We present alternative, q-hypergeometric proofs of some polynomial analogues of classical q-series identities recently discovered by Alladi and Berkovich, and Berkovich and Garvan.
This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with the same birthday?). It is shown that if the number of colors grows to infinity, the asymptotic distribution is either a Poisson mixture or a Normal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors. This result holds for any graph sequence, deterministic or random. On the other hand, when the n…
The addition-deletion theorems for hyperplane arrangements, which were originally shown in [H. Terao, Arrangements of hyperplanes and their freeness I, II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 293--320], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the Euler multiplicity, of a restricted multiarrangement. We compute the Euler multiplicities in many cases. Then we apply the addition-deletion theorems to various a…
The aim of this paper is to develop a theory of finite transformation monoids and in particular to study primitive transformation monoids. We introduce the notion of orbitals and orbital digraphs for transformation monoids and prove a monoid version of D. Higman's celebrated theorem characterizing primitivity in terms of connectedness of orbital digraphs. A thorough study of the module (or representation) associated to a transformation monoid is initiated. In particular, we compute the projective cover of the transformation module over a field of characteristic zero in the case of a transitive…
