Linked partition ideals and the Alladi--Schur theorem

TL;DR

This paper derives advanced generating functions for a specific class of integer partitions with difference and divisibility constraints, providing an analytic refinement of the Alladi--Schur theorem.

Contribution

It introduces new trivariate generating functions for partitions with difference and divisibility restrictions, extending Andrews' refinement of the Alladi--Schur theorem.

Findings

Derived explicit trivariate generating functions for the partition set

Extended Andrews' refinement of the Alladi--Schur theorem analytically

Provided combinatorial interpretations for the generating functions

Abstract

Let $\mathscr{S}$ denote the set of integer partitions into parts that differ by at least $3$, with the added constraint that no two consecutive multiples of $3$ occur as parts. We derive trivariate generating functions of Andrews--Gordon type for partitions in $\mathscr{S}$ with both the number of parts and the number of even parts counted. In particular, we provide an analytic counterpart of Andrews' recent refinement of the Alladi--Schur theorem.