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On the Numbers of Orbits of Permutations under an Operator Related to Eulerian Numbers
Shinji Tanimoto
Department of Mathematics, Kochi Joshi University, Kochi 780-8515, Japan
tanimoto@cc.kochi-wu.ac.jp
Annals of Combinatorics 8 (2) p.239-250 June, 2004
AMS Subject Classification:05A05, 05A10
Abstract:
In two previous papers an operator on permutations was introduced and its applications to Eulerian numbers were discussed by means of periods and orbits under the operator. In this paper, observing particular subsequences of permutations, an explicit formula for the number of orbits is given for each period. Several identities concerning the number of orbits and its related numbers are also derived.
Keywords: permutations, Eulerian numbers, orbits under an action

References:

1. R.L. Graham, D.E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 1989.

2. A. Kerber, Algebraic Combinatorics via Finite Group Actions, BI-Wissenschaftsverlag, Mannheim, 1991.

3. D.E. Knuth, The Art of Computer Programming, Vol. 3, Sorting and Searching, Addison- Wesley, Reading, 1973.

4. L. Lesieur and J.-L. Nicolas, On the Eulerian numbers Mn = max1≤k≤n A(n;k), Europ. J. Combin. 13 (1992) 379--399.

5. J.E.A. Steggall, On the numbers of patterns which can be derived from certain elements, Messenger Math. 37 (1907) 56--61.

6. S. Tanimoto, An operator on permutations and its application to Eulerian numbers, Europ. J. Combin. 22 (2001) 569--576.

7. S. Tanimoto, A study of Eulerian numbers by means of an operator on permutations, Europ. J. Combin. 24 (2003) 33--43.

8. J.H. van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, 1992.