Binomial coefficients and multifactorial numbers through generative grammars

TL;DR

This paper explores the use of formal derivatives in context-free and matrix grammars to analyze and generate binomial coefficients and multifactorial numbers, providing new methods for their mathematical representation.

Contribution

It introduces an extension of the formal derivative operator to matrix grammars, enabling the generation of multifactorial numbers and offering novel insights into their properties.

Findings

Formal derivatives can be used to prove properties of binomial coefficients.

Multifactorial numbers can be generated using extended matrix grammars.

The approach provides a new framework for understanding combinatorial numbers.

Abstract

In this paper, the formal derivative operator defined with respect to context-free grammars is used to prove some properties about binomial coefficients and multifactorial numbers. In addition, we extend the formal derivative operator to matrix grammars and show that multifactorial numbers can also be generated.