An Andrews-Gordon Type Identity Related to Andrews' Parity Consideration

Andrews investigated parity conditions in the Rogers-Ramanujan-Gordon theorem. Under the conditions that even parts or odd parts appear an even number of times, Andrews discovered two Rogers-Ramanujan-Gordon type partition theorems and derived corresponding generating functions. In the Rogers-Ramanujan-Gordon theorem, there are two parameters $k$ and $a$, where $k-1$ is the maximum number of consecutive parts $l$ and $l+1$, and $a-1$ is the maximum number of parts equal to $1$. Andrews' first theorem deals with the case $k\equiv a \;(\rm{mod}\;2)$, while the second theorem concerns the case where $k$ is even and $a$ is odd. These two partition identities have different infinite product forms on the right-hand side. In this paper, we consider the case $k\not\equiv a \;(\rm{mod}\;2)$ and use Bailey's lemma to obtain an Andrews-Gordon type identity whose right-hand side coincides with that of Andrews' identity for the case $k\equiv a \;(\rm{mod}\;2)$. We were unable to find a suitable combinatorial interpretation of the infinite sum form of this expression in terms of partitions, but with the help of lattice paths, we provide an appropriate combinatorial interpretation.