Matricial formulae for partitions

TL;DR

This paper presents a matrix-based formula linking the exponential of a specific triangular matrix to the matrix of partition numbers, revealing a new algebraic structure for partition functions.

Contribution

It introduces a novel matricial formula connecting exponential matrices with partition numbers, generalizing to sequences satisfying particular recurrences.

Findings

Matrix exponential yields partition number matrix

Generalizes to sequences with special recurrences

Provides algebraic insights into partition functions

Abstract

The exponential of the triangular matrix whose entries in the diagonal at distance $n$ from the principal diagonal are all equal to the sum of the inverse of the divisors of $n$ is the triangular matrix whose entries in the diagonal at distance $n$ from the principal diagonal are all equal to the number of partitions of $n$. A similar result is true for any pair of sequences satisfying a special recurrence.