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Langford sequences: perfect and hooked

โœ Scribed by James E. Simpson

Publisher
Elsevier Science
Year
1983
Tongue
English
Weight
562 KB
Volume
44
Category
Article
ISSN
0012-365X

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โœฆ Synopsis

Asequence{d,d+l...., d f m -1) of m consecutive positive integers is said to be perfect if the integers {1,2, . . . , 2m} can be arranged in disjoint pairs {(q, bi): 1 si G m} so that {bi-a,: l~i~m}={d,d+l,..., d+m-1). A sequence is hooked if the set {1,2,...,2m-1,2m + 1) can be arranged in pairs to satisfy the same condition. Well known necessary conditions for perfect sequences are herein shown to be sufficient. Similar necessary and sufficient conditions for hooked sequences are given.

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We explore the relations between Langford (2, m, 3m)-sequences on the one hand and complete and addtive permutations on the other. We consider in this context permutations with a certain "splitting" property and report on the results of some computer studies.