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

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