The $(k,l)$-Euler theorem and the combinatorics of $(k,l)$-sequences

In 1997, Bousquet-M\'elou and Eriksson stated a broad generalization of Euler's distinct-odd partition theorem, namely the $(k,l)$-Euler theorem. Their identity involved the $(k,l)$-lecture-hall partitions, which, unlike usual difference conditions of partitions in Rogers-Ramanujan type identities, satisfy some ratio constraints. In a 2008 paper, in response to a question suggested by Richard Stanley, Savage and Yee provided a simple bijection for the $l$-lecture-hall partitions (the case $k=l$), whose specialization in $l=2$ corresponds to Sylvester's bijection. Subsequently, as an open question, a generalization of their bijection was suggested for the case $k,l\geq 2$. In the spirit of Savage and Yee's work, we provide and prove in this paper slight variations of the suggested bijection, not only for the case $k,l\geq 2$ but also for the cases $(k,1)$ and $(1,k)$ with $k\geq 4$. Furthermore, we show that our bijections equal the recursive bijections given by Bousquet-M\'elou and Eriksson in their recursive proof of the $(k,l)$-lecture hall and finally provide the analogous recursive bijection for the $(k,l)$-Euler theorem.