Related papers: Recombination and bitsets
Stern's diatomic sequence is a well-studied and simply defined sequence with many fascinating characteristics. The binary signed-digit representation of integers is an alternative representation of integers with much use in efficient…
Encodings, that is, injective functions from words to words, have been studied extensively in several settings. In computability theory the notion of encoding is crucial for defining computability on arbitrary domains, as well as for…
Packing density is a permutation occurrence statistic which describes the maximal number of permutations of a given type that can occur in another permutation. In this article we focus on containment of sets of permutations. Although this…
In this paper, we introduce the notion of recurrence entropy in the context of nonlinear control systems. A set is said to be ($\tau$-)recurrent if every trajectory that starts in the set returns to it (within at most $\tau$ units of time).…
This report presents PixelBytes, an approach for unified multimodal representation learning. Drawing inspiration from sequence models like Image Transformers, PixelCNN, and Mamba-Bytes, we explore integrating text, audio, action-state, and…
In the evolution of a genome, the gene sequence is sometimes rearranged, for example by transposition of two adjacent gene blocks. In biocombinatorics, one tries to reconstruct these rearrangement incidents from the resulting permutation.…
We introduce the notion of evolution on sets and study several sets endowed with this structure and obtain some results about this new notion.
Life and language are discrete combinatorial systems (DCSs) in which the basic building blocks are finite sets of elementary units: nucleotides or codons in a DNA sequence and letters or words in a language. Different combinations of these…
A number system coding for the permutations generated by cyclic shift is described. The system allows to find the rank of a permutation given how it has been generated, and to determine a permutation given its rank. It defines a code…
The task of learning to map an input set onto a permuted sequence of its elements is challenging for neural networks. Set-to-sequence problems occur in natural language processing, computer vision and structure prediction, where…
A permutation is defined to be cycle-up-down if it is a product of cycles that, when written starting with their smallest element, have an up-down pattern. We prove bijectively and analytically that these permutations are enumerated by the…
A generic computation of a subset $A$ of $\mathbb{N}$ is a computation which correctly computes most of the bits of $A$, but which potentially does not halt on all inputs. The motivation for this concept is derived from complexity theory,…
This paper introduces the notion of multiset codes as relevant to the problem of reliable information transmission over permutation channels. The motivation for studying permutation channels comes from the effect of out of order delivery of…
The prevalent approach to sequence to sequence learning maps an input sequence to a variable length output sequence via recurrent neural networks. We introduce an architecture based entirely on convolutional neural networks. Compared to…
This article describes lossless compression algorithms for multisets of sequences, taking advantage of the multiset's unordered structure. Multisets are a generalisation of sets where members are allowed to occur multiple times. A multiset…
The evolution in coding DNA sequences brings new flexibility and freedom to the codon words, even as the underlying nucleotides get significantly ordered. These curious contra-rules of gene organisation are observed from the distribution of…
De novo assembly is the process of reconstructing the genome sequence of an organism from sequencing reads. Genome sequences are essential to biology, and assembly has been a central problem in bioinformatics for four decades. Until…
In this paper we construct families of bit sequences using combinatorial methods. Each sequence is derived by con- verting a collection of numbers encoding certain combinatorial nu- merics from objects exhibiting symmetry in various…
Different ways to describe a permutation, as a sequence of integers, or a product of Coxeter generators, or a tree, give different choices to define a simple permutation. We recollect few of them, define new types of simple permutations,…
Pedigrees, or family trees, are graphs of family relationships that are used to study inheritance. A fundamental problem in computational biology is to find, for a pedigree with $n$ individuals genotyped at every site, a set of…