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

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 $k\ge 2$. 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.