# The Combinatorics of Weighted Vector Compositions

## TL;DR

This paper explores the combinatorial structure of weighted vector compositions, connecting them to probability, calculus, number theory, and asymptotic analysis, and proposing a new primality conjecture.

## Contribution

It introduces the concept of weighted vector compositions, linking them to various mathematical areas and extending classical results with new conjectures.

## Key findings

- Relations to sums of random vectors
- Formulas for derivatives of composite functions
- Congruence properties similar to binomial coefficients

## Abstract

A vector composition of a vector $\mathbf{\ell}$ is a matrix $\mathbf{A}$ whose rows sum to $\mathbf{\ell}$. We define a weighted vector composition as a vector composition in which the column values of $\mathbf{A}$ may appear in different colors. We study vector compositions from different viewpoints: (1) We show how they are related to sums of random vectors and (2) how they allow to derive formulas for partial derivatives of composite functions. (3) We study congruence properties of the number of weighted vector compositions, for fixed and arbitrary number of parts, many of which are analogous to those of ordinary binomial coefficients and related quantities. Via the Central Limit Theorem and their multivariate generating functions, (4) we also investigate the asymptotic behavior of several special cases of numbers of weighted vector compositions. Finally, (5) we conjecture an extension of a primality criterion due to Mann and Shanks in the context of weighted vector compositions.