Matrix representation for combinatorics
Many combinatorial quantities have natural representations in terms of denumerably infinite matrices. For example, the matrix \(B=(B_{ij})\), \(i,j=0,1,2,..\). where \(B_{ij}=\left( \begin{matrix} i\\ j\end{matrix} \right)\) represents the binomial coefficients, and if \(b_ k\) is the k-th Bernoulli number the infinite column vector \(b=(b_ k)\), \(k=0,1,2,..\). represents the Bernoulli numbers. By exploiting this idea together with some algebra of denumerably matrices, the author provides a simplification and unification of some traditional results as well as a method of discovery and proof of combinatorial identities. The key concept to his approach is that of the derivative of a denumerably infinite matrix.
representation of combinatorial quantities
scientific article; zbMATH DE number 3873333