The [formula omitted]-Euler theorem and the combinatorics of [formula omitted]-sequences

Abstract

In 1997, Bousquet-Mélou 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≥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≥2, but also for the cases (k,1) and (1,k) with k≥4. Furthermore, we show that our bijections equal the recursive bijections given by Bousquet-Mélou and Eriksson in their recursive proof of the (k,l)-lecture hall theorem and finally provide the analogous recursive bijection for the (k,l)-Euler theorem.