TY - JOUR
T1 - Euler's partition theorem and the combinatorics of ℓ-sequences
AU - Savage, Carla D.
AU - Yee, Ae Ja
N1 - Funding Information:
E-mail addresses: [email protected] (C.D. Savage), [email protected] (A.J. Yee). 1 Research supported in part by NSF grant DMS-0300034. 2 The author is an Alfred P. Sloan Research Fellow.
PY - 2008/8
Y1 - 2008/8
N2 - 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.
AB - 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.
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U2 - 10.1016/j.jcta.2007.11.006
DO - 10.1016/j.jcta.2007.11.006
M3 - Article
AN - SCOPUS:44149115203
SN - 0097-3165
VL - 115
SP - 967
EP - 996
JO - Journal of Combinatorial Theory. Series A
JF - Journal of Combinatorial Theory. Series A
IS - 6
ER -