Related papers: The insertion encoding of Cayley permutations
Permutation codes have received a great attention due to various applications. For different applications, one needs permutation codes under different metrics. The generalized Cayley metric was introduced by Chee and Vu [4] and this metric…
We introduce a new row insertion algorithm on decreasing tableaux and increasing tableaux, generalizing Edelman-Greene (EG) row insertion. Our row insertion algorithm is a nontrivial variation of Hecke column insertion which generalizes EG…
We introduce and solve an infinite class of loop integrals which generalises the well-known ladder series. The integrals are described in terms of single-valued polylogarithmic functions which satisfy certain differential equations. The…
Modern statistical learning theory and deep learning characterize generalization primarily in terms of continuous capacity control (e.g., norm-based regularization, margin maximization, low-rank bias). While highly successful in continuous…
Indexed languages are a classical notion in formal language theory, which has attracted attention in recent decades due to its role in higher-order model checking: They are precisely the languages accepted by order-2 pushdown automata. The…
Cross-modal retrieval methods build a common representation space for samples from multiple modalities, typically from the vision and the language domains. For images and their captions, the multiplicity of the correspondences makes the…
Applying machine learning to mathematical terms and formulas requires a suitable representation of formulas that is adequate for AI methods. In this paper, we develop an encoding that allows for logical properties to be preserved and is…
We propose the Transcendental Encoding Conjecture for decision problems, which asserts that every language in complexity class P encodes to an algebraic real (possibly rational or algebraic irrational) under its binary characteristic…
Approaches to signal representation and coding theory have traditionally focused on how to best represent signals using parsimonious representations that incur the lowest possible distortion. Classical examples include linear and non-linear…
Many combinatorial proofs rely on induction. When these proofs are formulated in traditional language, they can be bulky and unmanageable. Coalgebras provide a language which can reduce reduce many inductive proofs in graded poset theory to…
We first introduce the percolation problems associated with the graph theoretical concepts of $(k,l)$-sparsity, and make contact with the physical concepts of ordinary and rigidity percolation. We then devise a renormalization…
We define a generalization of the Eulerian polynomials and the Eulerian numbers by considering a descent statistic on segmented permutations coming from the study of 2-species exclusion processes and a change of basis in a Hopf algebra. We…
We collect open problems in permutation patterns on four themes: rank-unimodality in the permutation pattern poset, Wilf-equivalence and shape-Wilf-equivalence, the enumeration of derangements in permutation classes, and sorting by stacks…
We consider the embedding problem in coding theory: given an independence (a code-related property) and an independent language $L$, find a maximal independent language containing $L$. We consider the case where the code-related property is…
We introduce a general Bayesian framework for graph matching grounded in a new theory of exchangeable random permutations. Leveraging the cycle representation of permutations and the literature on exchangeable random partitions, we define,…
By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial…
By introducing Hilbert space and operators, we show how probabilities, approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates. Our main results yield orthogonal bases which…
We conservatively extend classical elementary differential calculus to the Cartesian closed category of convergence spaces. By specializing results about the convergence space representation of directed graphs, we use Cayley graphs to…
In this paper, we introduce a new way of constructing and decoding multipermutation codes. Multipermutations are permutations of a multiset that generally consist of duplicate entries. We first introduce a class of binary matrices called…
Recursive coalgebras provide an elegant categorical tool for modelling recursive algorithms and analysing their termination and correctness. By considering coalgebras over categories of suitably indexed families, the correctness of the…