Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-25T04:57:31.079Z Has data issue: false hasContentIssue false
  • English
  • Français

Permutational labelling of constant weight Gray codes

Published online by Cambridge University Press:  17 April 2009

Inessa Levi  and
Steve Seif
Inessa Levi
Affiliation:
Department of Mathematics, University of Louisville, Louisville, KY 40292, United States of America
Steve Seif
Affiliation:
Department of Mathematics, University of Louisville, Louisville, KY 40292, United States of America
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that for positive integers n and r satisfying 1 < r < n, with the single exception of n = 4 and r = 2, there exists a constant weight Gray code of r-sets of Xn = {1, 2, …, n} that admits an orthogonal labelling by distinct partitions, with each subsequent partition obtained from the previous one by an application of a permutation of the underlying set. Specifically, an r-set A and a partition π of Xn are said to be orthogonal if every class of π meets A in exactly one element. We prove that for all n and r as stated, and taken modulo , there exists a list of the distinct r-sets of Xn with |Ai ∩ Ai+1| = r − 1 and a list of distinct partitions such that πi is orthogonal to both Ai and Ai+1, and πi+1 = πiλi for a suitable permutation λi of Xn.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 3 , June 2002 , pp. 399 - 406
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Howie, J.M. and McFadden, R.B., ‘Idempotent rank in finite full transformation semigroups’, Proc. Royal Soc. Edinburgh 114 (1990), 161–167.CrossRefGoogle Scholar
[2]Lehel, J. and Levi, I., ‘Loops with partitions and matchings’, Ars Combin. 54 (2000), 237–253.Google Scholar
[3]Levi, I., McFadden, R.B. and Seif, S., ‘Algorithms for labelling Gray codes’, (submitted).Google Scholar
[4]Levi, I. and Seif, S., ‘Constant weight Gray codes labeled by partitions with blocks of size at most two’, Ars Combin. (to appear).Google Scholar
[5]Reingold, E.M., Nievergelt, J. and Deo, N., Combinatorial algorithms, theory and practice (Prentice-Hall, Englewood Cliffs, NJ, 1977).Google Scholar
[6]Savage, C.D. and Winkler, P., ‘Monotone Gray codes and the middle levels problem’, J. Combin. Theory Ser. A 70 (1995), 230–248.CrossRefGoogle Scholar
[7]Wilf, H.S., Combinatorial algorithms: an update,CBMS-NSF Regional Conference Series in Applied Mathematics 55 (SIAM, Philadelphia, PA, 1989).CrossRefGoogle Scholar
[8]van Zanten, A.J., ‘Index system and separability of constant weight Gray codes’, IEEE Trans. Inform. Theory 37 (1991), 1229–1223.CrossRefGoogle Scholar