Euler’s partition theorem and the combinatorics of `-sequences

Euler’s partition theorem states that the number of partitions of an integer N into odd parts is equal to the number of partitions of N in which the ratio of successive parts is greater than 1. It was shown by Bousquet-Mélou and Eriksson in [9] that a similar result holds when “odd parts” is replaced by “parts that are sums of successive terms of an `-sequence” and the ratio “1” is replaced by a root of the characteristic polynomial of the `-sequence. This generalization of Euler’s theorem is intrinsically different from the many others that have appeared, as it involves a family of partitions constrained by the ratio of successive parts. In this paper, we provide a surprisingly simple bijection for this result, a question suggested by Richard Stanley. In fact, we give a parametrized family of bijections, that include, as special cases, Sylvester’s bijection and a bijection for the lecture hall theorem. We introduce Sylvester diagrams as a way to visualize these bijections and deduce their properties. In proving the bijections, we uncover the intrinsic role played by the combinatorics of `sequences and use this structure to give a combinatorial charaterization of the partitions defined by the ratio constraint. Several open questions suggested by this work are described.