Sylvester-Cayley vector partitions algorithm and the Gaussian polynomials

TL;DR

This paper extends a historical algorithm for double partitions to derive explicit formulas for Gaussian polynomial coefficients using convolution of restricted partition functions.

Contribution

It introduces a novel extension of Sylvester and Cayley's algorithm to compute Gaussian polynomial coefficients explicitly.

Findings

Derived explicit formulas for Gaussian polynomial coefficients.

Connected double partitions with convolution of restricted partition functions.

Extended classical algorithms to modern polynomial coefficient computation.

Abstract

We extend an algorithm suggested in 1858 by Sylvester and implemented in 1860 by Cayley for a problem of double partitions and apply it to derivation of explicit expressions for coefficients of the Gaussian polynomials through convolution of restricted partition functions.