Euler's partition theorem and the combinatorics of ℓ-sequences |
| |
Authors: | Carla D. Savage Ae Ja Yee |
| |
Affiliation: | aComputer Science Department, Box 8206, North Carolina State University, Raleigh, NC 27695-8206, USA;bDepartment of Mathematics, The Pennsylvania State University, 327 McAllister Building, University Park, PA 16802, USA |
| |
Abstract: | 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 [M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions II, Ramanujan J. 1 (2) (1997) 165–185] that a similar result holds when “odd parts” is replaced by “parts that are sums of successive terms of an ℓ-sequence” and the ratio “1” is replaced by a root of the characteristic polynomial of the ℓ-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 ℓ-sequences and use this structure to give a combinatorial characterization of the partitions defined by the ratio constraint. Several open questions suggested by this work are described. |
| |
Keywords: | Integer partitions Lecture hall partitions Euler's partition theorem Sylvester's bijection Partition bijections |
本文献已被 ScienceDirect 等数据库收录! |
|