Abstract
A model for linearly ordering basic classes of combinatorial sets is developed in terms of chains of partitions. In this context general procedures for locating the position of an object given the object (ranking) and for constructing an object given its position (unranking) are described. A general method of associating a labeled tree with a chain of partitions together with a reduction operation producing classes of labeled graphs from trees is presented. These latter operations relate these ideas to a general setting for sequencing, ranking, and selection algorithms due to H. S. Wilf.
Keywords
- Linear Order
- Label Graph
- Label Tree
- Combinatorial Object
- Natural Identification
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Research supported by N. S. F. Grant MCS 74-02714-A02
Author at present on leave to University of Minnesota, Minneapolis, Minnesota, 55455, from University of California, San Diego, La Jolla, California, 92093.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Calabi, E. and Wilf, H., On the sequential and random selection of subspaces over a finite field, J. Combinatorial Theory, to appear.
Ehrlich, G. Loopless algorithms for generating permutations and other combinatorial configurations, J. Assoc. for Computing Mach., V. 20, No. 3 (1973) pp. 500–513.
Even, S., Algorithmic Combinatorics, Macmillan, New York, 1973.
Fillmore, J. and Williamson S. G., On backtracking: a combinatorial description of the algorithm, SIAM J. Comput., V. 3, No. 1(March 1974) pp. 41–55.
Fillmore, J. and Williamson S. G., On ranking functions: the symmetries and colorations of the n-cube, SIAM J. Comput. (to appear).
Foata, D. and Schutzenberger, M.P., Theorie des polynômes eulériens, Lecture Notes in Math., No. 138, Springer Verlag, Berlin (1970).
Ives, F. M., Permutation enumeration: four new permutation algorithms, Communications ACM, v. 19, no. 2(Feb. 1976), pp. 68–72.
Joichi, J., White D., and Williamson S. G., Linear space constant time orders for set partitions (manuscript in preparation, U. of Minn.)
Knuth, D. E., Sorting and Searching, The Art of Computer Programming, V. 3, Addison Wesley, 1973, p. 46 and 587.
Lehmer, D. H., The machine tools of combinatorics, in Applied Combinatorial Mathematics, E. F. Beckenbach (ed), Wiley, New York, 1964, pp. 5–31.
Milne, S., Restricted growth functions and incidence relations in the lattice of partitions (preprint Univ. of Calif., San Diego).
Wilf, H.S., A unified setting for sequencing, ranking, and selection algorithms for combinatorial objects (preprint Univ. of Pennsylvania).
Williamson, S. G., Ranking algorithms for lists of partitions, SIAM J. Comput. (to appear).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1977 Springer-Verlag
About this paper
Cite this paper
Williamson, S.G. (1977). On the ordering, ranking, and random generation of basic combinatorial sets. In: Foata, D. (eds) Combinatoire et Représentation du Groupe Symétrique. Lecture Notes in Mathematics, vol 579. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090026
Download citation
DOI: https://doi.org/10.1007/BFb0090026
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-08143-2
Online ISBN: 978-3-540-37385-8
eBook Packages: Springer Book Archive
