Related papers: The Joseph Greenberg problem: combinatorics and co…

The problem of combining the evidence concerning an unknown, contained in each of $k$ Bayesian inference bases, is discussed. This can be considered as a generalization of the problem of pooling $k$ priors to determine a consensus prior.…

The sequence of partial sums of Fibonacci numbers, beginning with $2$, $4$, $7$, $12$, $20$, $33,\dots$, has several combinatorial interpretations (OEIS A000071). For instance, the $n$-th term in this sequence is the number of length-$n$…

Much progress has been made on the problem of calculating $g(n)$ for various classes of integers $n$, where $g$ is the group-counting function. We approach the inverse problem of solving the equations $g(n) = 6$ and $g(n) = 7$ in $n$. The…

In a classic paper Schr\"oder posed four combinatorial problems about the number of certain types of bracketings of words and sets. Here we address what these bracketings look like on average. For each of the four problems we prove that a…

In 1981, Andr\'e Joyal provided a combinatorial interpretation of the algebra of formal power series, a central gadget in the toolkit of enumerative combinatorics. In Joyal's theory of species of structures, combinatorial species (like…

Speech-comprehension difficulties are common among older people. Standard speech tests do not fully capture such difficulties because the tests poorly resemble the context-rich, story-like nature of ongoing conversation and are typically…

An archetypal problem discussed in computer science is the problem of searching for a given number in a given set of numbers. Other than sequential search, the classic solution is to sort the list of numbers and then apply binary search.…

Natural language processing researchers have identified limitations of evaluation methodology for generation tasks, with new questions raised about the validity of automatic metrics and of crowdworker judgments. Meanwhile, efforts to…

Many tasks in Natural Language Processing involve recognizing lexical entailment. Two different approaches to this problem have been proposed recently that are quite different from each other. The first is an asymmetric similarity measure…

Cross-entropy loss is a common choice when it comes to multiclass classification tasks and language modeling in particular. Minimizing this loss results in language models of very good quality. We show that it is possible to fine-tune these…

We use a probabilistic method to produce some combinatorial inequalities by considering pattern containment in permutations and words.

This article aims to be a self-contained account of the history of the B. B. Newman Spelling Theorem, including the historical context in which it arose. First, an account of B. B. Newman and how he came to prove his Spelling Theorem is…

We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on words. The extension is based on…

In this paper we look at the ability of recent large language models (LLMs) at solving mathematical problems in combinatorics. We compare models LLaMA-2, LLaMA-3.1, GPT-4, and Mixtral against each other and against human pupils and…

In this paper we propose an approach to implement specific relation-ship set between two entities called combinatorial relationship set. For the combinatorial relationship set B between entity sets G and I the mapping cardinality is…

We describe an implementation of a genetic algorithm on partially commutative groups and apply it to the double coset search problem on a subclass of groups. This transforms a combinatorial group theory problem to a problem of combinatorial…

A quantitative method is suggested, where meanings of words, and grammatic rules about these, of a vocabulary are represented by real numbers. People meet randomly, and average their vocabularies if they are equal; otherwise they either…

Let $A$ be a finite set of relatively prime positive integers, and let $S(A)$ be the set of all nonnegative integral linear combinations of elements of $A$. The set $S(A)$ is a semigroup that contains all sufficiently large integers. The…

Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely…

The notion of the Fibonacci cobweb poset from [1] has been naturally extended to any admissible sequence $F$ in [2] where it was also recognized that the celebrated prefab notion of Bender and Goldman [3] - (see also [4,5]) - admits such an…