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We consider linear error correcting codes associated to higher dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult questions in combinatorics and algebraic geometry. This is illustrated by codes associated to Schubert varieties in Grassmannians, called Schubert codes, which have recently been studied. The basic parameters such as the length, dimension and minimum distance of these codes are known only in special cases. An upper bound for the minimum distance is known and it is conjectured that this bound is achieved. We give explicit formulae for the length and dimension of arbitrary Schubert codes and prove the minimum distance conjecture in the affirmative for codes associated to Schubert divisors.

We survey some general-purpose symbolic software packages that implement algorithms from enumerative and analytic combinatorics. Software for the following areas is covered: basic combinatorial objects, symbolic combinatorics, P\'olya theory, combinatorial species, and asymptotics. We describe the capabilities that the packages offer as well as some of the algorithms used, and provide links to original documentation. Most of the packages are freely downloadable from the web.

A survey written for the upcoming "Handbook of Enumerative Combinatorics".

This article is from Journal of Cheminformatics , volume 5 . Abstract Background: There is a growing body of literature describing the properties of marketed drugs, the concept of drug-likeness and the vastness of chemical space. In that context, enumerative combinatorics with simple atomic components may be useful in the conception and design of structurally novel compounds for expanding and enhancing high-throughput screening (HTS) libraries. Results: A random combination of mono- and diatomic carbon, hydrogen, nitrogen, and oxygen containing components in the absence of molecular weight constraints but with the ability to form rings affords virtual compounds that fall in bulk physicochemical space typically associated with drugs, but whose ring assemblies fall in new or under-represented areas of chemical shape space. When compared against compounds in the ChEMBL_14, MDDR, Drug Bank and Dictionary of Natural Products, the percentage of virtual compounds with a Tanimoto index of 1.0 (ECFP_4) was found to be as high as 0.21. Depending on therapeutic target, this value may be in range of what might be expected from an experimental HTS campaign in terms of a true hit rate. Conclusion: Virtual compounds derived through enumerative combinatorics of simple atomic components have drug-like properties with ring assemblies that fall in new or under-represented areas of shape space. Structures derived in this manner could provide the starting point or inspiration for the design of structurally novel scaffolds in an unbiased fashion.

This article is from Journal of Cheminformatics , volume 5 . Abstract Background: There is a growing body of literature describing the properties of marketed drugs, the concept of drug-likeness and the vastness of chemical space. In that context, enumerative combinatorics with simple atomic components may be useful in the conception and design of structurally novel compounds for expanding and enhancing high-throughput screening (HTS) libraries. Results: A random combination of mono- and diatomic carbon, hydrogen, nitrogen, and oxygen containing components in the absence of molecular weight constraints but with the ability to form rings affords virtual compounds that fall in bulk physicochemical space typically associated with drugs, but whose ring assemblies fall in new or under-represented areas of chemical shape space. When compared against compounds in the ChEMBL_14, MDDR, Drug Bank and Dictionary of Natural Products, the percentage of virtual compounds with a Tanimoto index of 1.0 (ECFP_4) was found to be as high as 0.21. Depending on therapeutic target, this value may be in range of what might be expected from an experimental HTS campaign in terms of a true hit rate. Conclusion: Virtual compounds derived through enumerative combinatorics of simple atomic components have drug-like properties with ring assemblies that fall in new or under-represented areas of shape space. Structures derived in this manner could provide the starting point or inspiration for the design of structurally novel scaffolds in an unbiased fashion.

This article is from Journal of Cheminformatics , volume 5 . Abstract Background: There is a growing body of literature describing the properties of marketed drugs, the concept of drug-likeness and the vastness of chemical space. In that context, enumerative combinatorics with simple atomic components may be useful in the conception and design of structurally novel compounds for expanding and enhancing high-throughput screening (HTS) libraries. Results: A random combination of mono- and diatomic carbon, hydrogen, nitrogen, and oxygen containing components in the absence of molecular weight constraints but with the ability to form rings affords virtual compounds that fall in bulk physicochemical space typically associated with drugs, but whose ring assemblies fall in new or under-represented areas of chemical shape space. When compared against compounds in the ChEMBL_14, MDDR, Drug Bank and Dictionary of Natural Products, the percentage of virtual compounds with a Tanimoto index of 1.0 (ECFP_4) was found to be as high as 0.21. Depending on therapeutic target, this value may be in range of what might be expected from an experimental HTS campaign in terms of a true hit rate. Conclusion: Virtual compounds derived through enumerative combinatorics of simple atomic components have drug-like properties with ring assemblies that fall in new or under-represented areas of shape space. Structures derived in this manner could provide the starting point or inspiration for the design of structurally novel scaffolds in an unbiased fashion.

In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in combinatorics, Ehrhart's method for proving that a couting function is a polynomial, the connection between polyhedral cones, rational functions and quasisymmetric functions, methods for bounding coefficients, combinatorial reciprocity theorems, algorithms for counting integer points in polyhedra and computing rational function representations, as well as visualizations of the greatest common divisor and the Euclidean algorithm.

We recall that the full susceptibility series of the Ising model, modulo powers of the prime 2, reduce to algebraic functions. We also recall the non-linear polynomial differential equation obtained by Tutte for the generating function of the q-coloured rooted triangulations by vertices, which is known to have algebraic solutions for all the numbers of the form $2 +2 \cos(j\pi/n)$, the holonomic status of the q= 4 being unclear. We focus on the analysis of the q= 4 case, showing that the corresponding series is quite certainly non-holonomic. Along the line of a previous work on the susceptibility of the Ising model, we consider this q=4 series modulo the first eight primes 2, 3, ... 19, and show that this (probably non-holonomic) function reduces, modulo these primes, to algebraic functions. We conjecture that this probably non-holonomic function reduces to algebraic functions modulo (almost) every prime, or power of prime numbers. This raises the question to see whether such remarkable non-holonomic functions can be seen as ratio of diagonals of rational functions, or algebraic, functions of diagonals of rational functions.

We recall that the full susceptibility series of the Ising model, modulo powers of the prime 2, reduce to algebraic functions. We also recall the non-linear polynomial differential equation obtained by Tutte for the generating function of the q-coloured rooted triangulations by vertices, which is known to have algebraic solutions for all the numbers of the form $2 +2 \cos(j\pi/n)$, the holonomic status of the q= 4 being unclear. We focus on the analysis of the q= 4 case, showing that the corresponding series is quite certainly non-holonomic. Along the line of a previous work on the susceptibility of the Ising model, we consider this q=4 series modulo the first eight primes 2, 3, ... 19, and show that this (probably non-holonomic) function reduces, modulo these primes, to algebraic functions. We conjecture that this probably non-holonomic function reduces to algebraic functions modulo (almost) every prime, or power of prime numbers. This raises the question to see whether such remarkable non-holonomic functions can be seen as ratio of diagonals of rational functions, or algebraic, functions of diagonals of rational functions.

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 will try to sketch Professor F. Y. Wu's contributions in lattice statistical mechanics, solid state physics, graph theory, enumerative combinatorics and so many other domains of physics and mathematics. We will recall F. Y. Wu's most important and well-known classic results and we will also sketch his most recent researches dedicated to the connections of lattice statistical mechanical models with deep problems in pure mathematics. Since it is hard to provide an exhaustive list of all his contributions, to give some representation of F. Y. Wu's "mental connectivity" we will concentrate on the interrelations between the various results he has obtained in so many different domains of physics and mathematics. Along the way we will also try to understand Wu's motivations and his favorite concepts, tools and ideas.

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 distributions which are $\textit{perturbed log-concave}$, via a quantitative local central limit theorem. We then use it to generalize some classical asymptotic statistics in combinatorial theory, as well as assert some new ones.

We construct a family of weight functions on finite abelian groups that yield invertible MacWilliams identities for additive codes. The weights are obtained composing a suitable support map with the rank function of a graded lattice that satisfies certain regularity properties. We express the Krawtchouk coefficients of the corresponding MacWilliams transformation in terms of the combinatorial invariants of the underlying lattice, and show that the most relevant weight functions studied in coding theory belong, up to equivalence, to the class that we introduce. In particular, we compute some classical Krawtchouk coefficients employing a simple combinatorial method. Our approach also allows to systematically construct weight functions that endow the underlying group with a metric space structure. We establish a Singleton-like bound for additive codes, and call optimal the codes that attain the bound. Then we prove that the dual of an optimal code is optimal, and that the weight distribution of an optimal code is completely determined by three fundamental parameters. Finally, we apply MacWilliams identities for the rank weight to enumerative combinatorics problems, computing the number of matrices of given rank over a finite field that satisfy certain linear conditions.

We construct a family of weight functions on finite abelian groups that yield invertible MacWilliams identities for additive codes. The weights are obtained composing a suitable support map with the rank function of a graded lattice that satisfies certain regularity properties. We express the Krawtchouk coefficients of the corresponding MacWilliams transformation in terms of the combinatorial invariants of the underlying lattice, and show that the most relevant weight functions studied in coding theory belong, up to equivalence, to the class that we introduce. In particular, we compute some classical Krawtchouk coefficients employing a simple combinatorial method. Our approach also allows to systematically construct weight functions that endow the underlying group with a metric space structure. We establish a Singleton-like bound for additive codes, and call optimal the codes that attain the bound. Then we prove that the dual of an optimal code is optimal, and that the weight distribution of an optimal code is completely determined by three fundamental parameters. Finally, we apply MacWilliams identities for the rank weight to enumerative combinatorics problems, computing the number of matrices of given rank over a finite field that satisfy certain linear conditions.