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Register a new userOn the Algebraic Combinatorics of Injections and its Applications to Injection Codes
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We consider the algebraic combinatorics of the set of injections from a $k$-element set to an $n$-element set. In particular, we give a new combinatorial formula for the spherical functions of the Gelfand pair $(S_k times S_n, text{diag}(S_k) times S_{n-k})$. We use this combinatorial formula to give new Delsarte linear programming bounds on the size of codes over injections.
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The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singletons defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as extremal codes, Hermitian codes, MDS and NMDS codes. Moreover, we are able to present for the first time the weight distribution of AMDS codes. We also discuss the link between our results and the Pless equations.
We obtain a necessary and sufficient condition for the linear independence of solutions of differential equations for hyperlogarithms. The key fact is that the multiplier (i.e. the factor $M$ in the differential equation $dS=MS$) has only singularities of first order (Fuchsian-type equations) and this implies that they freely span a space which contains no primitive. We give direct applications where we extend the property of linear independence to the largest known ring of coefficients.
Background: We study the sparsification of dynamic programming folding algorithms of RNA structures. Sparsification applies to the mfe-folding of RNA structures and can lead to a significant reduction of time complexity. Results: We analyze the sparsification of a particular decomposition rule, $Lambda^*$, that splits an interval for RNA secondary and pseudoknot structures of fixed topological genus. Essential for quantifying the sparsification is the size of its so called candidate set. We present a combinatorial framework which allows by means of probabilities of irreducible substructures to obtain the expected size of the set of $Lambda^*$-candidates. We compute these expectations for arc-based energy models via energy-filtered generating functions (GF) for RNA secondary structures as well as RNA pseudoknot structures. For RNA secondary structures we also consider a simplified loop-energy model. This combinatorial analysis is then compared to the expected number of $Lambda^*$-candidates obtained from folding mfe-structures. In case of the mfe-folding of RNA secondary structures with a simplified loop energy model our results imply that sparsification provides a reduction of time complexity by a constant factor of 91% (theory) versus a 96% reduction (experiment). For the full loop-energy model there is a reduction of 98% (experiment).
One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. It is easily seen that the Hoffman bound is sharp on the tensor power of a graph whenever it is sharp for the original graph. In this paper, we introduce the related problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many of the prominent open problems in extremal combinatorics, such as the Turan problem for (hyper-)graphs, can be encoded as special cases of this problem. We also give a new generalization of the Hoffman bound for hypergraphs which is sharp for the tensor power of a hypergraph whenever it is sharp for the original hypergraph. As an application of our Hoffman bound, we make progress on the problem of Frankl on families of sets without extended triangles from 1990. We show that if $frac{1}{2}nle2klefrac{2}{3}n,$ then the extremal family is the star, i.e. the family of all sets that contains a given element. This covers the entire range in which the star is extremal. As another application, we provide spectral proofs for Mantels theorem on triangle-free graphs and for Frankl-Tokushige theorem on $k$-wise intersecting families.
Building upon the rule-algebraic stochastic mechanics framework, we present new results on the relationship of stochastic rewriting systems described in terms of continuous-time Markov chains, their embedded discrete-time Markov chains and certain types of generating function expressions in combinatorics. We introduce a number of generating function techniques that permit a novel form of static analysis for rewriting systems based upon marginalizing distributions over the states of the rewriting systems via pattern-counting observables.
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