Related papers: Prolific Compositions
Compositional generalization, the ability to recognize familiar parts in novel contexts, is a defining property of intelligent systems. Although modern models are trained on massive datasets, they still cover only a tiny fraction of the…
For a finite field of odd number of elements we construct families of permutation binomials and permutation trinomials with one fixed-point (namely zero) and remaining elements being permuted as disjoint cycles of same length. Binomials and…
For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…
Natural language allows us to refer to novel composite concepts by combining expressions denoting their parts according to systematic rules, a property known as \emph{compositionality}. In this paper, we study whether the language emerging…
We show that there exists a bounded pattern of m consecutive primes for any m>0, that means a tuple H_m of m distinct non-negative integers h_i (i=1,2,...m) such that its translations contain arbitrarily long (finite) arithmetic…
We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime $p$. In particular, in the integer case, we improve a recent bound…
A permutation $\pi$ contains a pattern $\sigma$ if and only if there is a subsequence in $\pi$ with its letters are in the same relative order as those in $\sigma$. Partially ordered patterns (POPs) provide a convenient way to denote…
A generally intelligent learner should generalize to more complex tasks than it has previously encountered, but the two common paradigms in machine learning -- either training a separate learner per task or training a single learner for all…
A structured approach for the Collatz conjecture is presented using just the odd integers that are, in turn, divided into categories based on the roles they play such as Starter, Intermediary and Terminal. The expression 4x+1 is used as a…
Spurious ambiguity is the phenomenon whereby distinct derivations in grammar may assign the same structural reading, resulting in redundancy in the parse search space and inefficiency in parsing. Understanding the problem depends on…
Computability theory is traditionally conceived as the theoretical basis of informatics. Nevertheless, numerous proposals transcend computability theory, in particular by emphasizing interaction of modules, or components, parts,…
We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that…
For a triple of complex hyperplane arrangements, there is a well-known long exact sequence relating the cohomology of the complements. We observe that this result extends to certain local coefficient systems, and use this extension to study…
We study pattern avoidance by combinatorial objects other than permutations, namely by ordered partitions of an integer and by permutations of a multiset. In the former case we determine the generating function explicitly, for integer…
Let $S$ be a finite set of positive integers with largest element $m$. Let us randomly select a composition $a$ of the integer $n$ with parts in $S$, and let $m(a)$ be the multiplicity of $m$ as a part of $a$. Let $0\leq r<q$ be integers,…
We apply a common measure of randomness, the entropy, in the context of iterated functions on a finite set with n elements. For a permutation, it turns out that this entropy is asymptotically (for a growing number of iterations) close to…
Structure is one of the most essential aspects of music, and music structure is commonly indicated through repetition. However, the nature of repetition and structure in music is still not well understood, especially in the context of music…
In this paper, we extend the work of Andrews, Beck and Hopkins by considering partitions and compositions with bounded gaps between each pair of consecutive parts. We show that both their generating functions and two matrices determined by…
We characterise those permutation classes whose simple permutations are monotone griddable. This characterisation is obtained by identifying a set of nine substructures, at least one of which must occur in any simple permutation containing…
Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate…