A Generalized Burge Correspondence and $k$-measure of Partitions

TL;DR

This paper generalizes Burge's partition-word correspondence to encode partitions with distinct parts over a k-ary alphabet, enabling refined identities involving a new k-measure and connecting to classical gap-size partition theorems.

Contribution

It extends Burge's correspondence to k-ary alphabets for partitions with distinct parts, providing new combinatorial encodings and refinements of recent algebraic partition identities.

Findings

Established k-ary encodings for partitions with distinct parts.

Proved refined partition identities involving a new k-measure.

Connected the encoding to classical gap-size partition theorems.

Abstract

Let $P$ be the set of integer partitions and $D$ the subset of those with distinct parts. We extend a correspondence of Burge between partitions and binary words to give encodings of both $D$ and $D$ as words over a $k$-ary alphabet, for any fixed $k\geq 2$. These are used to prove refinements of two partition identities involving $k$-measure that were recently derived algebraically by Andrews, Chern and Li. The relationship between our encoding of $D$ and minimum gap-size partition identities (e.g. Schur's Theorem) is also briefly discussed.