Multiple partitions, lattice paths and a Burge-Bressoud-type correspondence

TL;DR

This paper introduces a bijection linking specific constrained partitions to ordered partitions and lattice paths, providing a new combinatorial proof of partition enumeration formulas and connecting to Bressoud's Burge correspondence.

Contribution

It presents a novel bijection between constrained partitions and ordered partitions with path interpretations, extending Bressoud's Burge correspondence.

Findings

Provides an elementary constructive proof of Andrews' multiple-sum partition enumeration

Establishes a natural relation between ordered partitions and restricted lattice paths

Extends Bressoud's Burge correspondence to new combinatorial structures

Abstract

A bijection is presented between (1): partitions with conditions $f_j+f_{j+1}\leq k-1$ and $ f_1\leq i-1$, where $f_j$ is the frequency of the part $j$ in the partition, and (2): sets of $k-1$ ordered partitions $(n^{(1)}, n^{(2)}, ..., n^{(k-1)})$ such that $n^{(j)}_\ell \geq n^{(j)}_{\ell+1} + 2j$ and $ n^{(j)}_{m_j} \geq j+ {\rm max} (j-i+1,0)+ 2j (m_{j+1}+... + m_{k-1})$, where $m_j$ is the number of parts in $n^{(j)}$. This bijection entails an elementary and constructive proof of the Andrews multiple-sum enumerating partitions with frequency conditions. A very natural relation between the $k-1$ ordered partitions and restricted paths is also presented, which reveals our bijection to be a modification of Bressoud's version of the Burge correspondence.