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Enumeration Schemes for Restricted Permutations

Published online by Cambridge University Press:  01 January 2008

VINCENT VATTER
VINCENT VATTER*
Affiliation:
School of Mathematics and Statistics, University of St AndrewsSt Andrews, Fife KY19 9SS, UK (e-mail: vince@mcs.st-and.ac.ukhttp://turnbull.mcs.st-and.ac.uk/~vince)
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Abstract

Zeilberger's enumeration schemes can be used to completely automate the enumeration of many permutation classes. We extend his enumeration schemes so that they apply to many more permutation classes and describe the Maple package WilfPlus, which implements this process. We also compare enumeration schemes to three other systematic enumeration techniques: generating trees, substitution decompositions, and the insertion encoding.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 1 , January 2008 , pp. 137 - 159
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Albert, M. H. and Atkinson, M. D. (2005) Simple permutations and pattern restricted permutations. Discrete Math. 300 1–15.Google Scholar
[2]Albert, M. H., Linton, S. and Ruškuc, N. (2005) The insertion encoding of permutations. Electron. J. Combin. 12 # 47 (electronic).Google Scholar
[3]Atkinson, M. D. (1998) Permutations which are the union of an increasing and a decreasing subsequence. Electron. J. Combin. 5 # 6 (electronic).CrossRefGoogle Scholar
[4]Atkinson, M. D. and Beals, R. (1999) Permuting mechanisms and closed classes of permutations. In Combinatorics, Computation & Logic '99 (Auckland), Vol. 21 of Aust. Comput. Sci. Commun., Springer, Singapore, pp. 117–127.Google Scholar
[5]Atkinson, M. D., Murphy, M. M. and Ruškuc, N. (2002) Partially well-ordered closed sets of permutations. Order 19 101–113.CrossRefGoogle Scholar
[6]Atkinson, M. D., Murphy, M. M. and Ruškuc, N. (2002) Sorting with two ordered stacks in series. Theoret. Comput. Sci. 289 205–223.CrossRefGoogle Scholar
[7]Atkinson, M. D., Murphy, M. M. and Ruškuc, N. (2005) Pattern avoidance classes and subpermutations. Electron. J. Combin. 12 # 60 (electronic).Google Scholar
[8]Barcucci, E., Del Lungo, A.Pergola, E. and Pinzani, R. (1999) ECO: A methodology for the enumeration of combinatorial objects. J. Differ. Equations Appl. 5 435–490.Google Scholar
[9]Billey, S. C. and Warrington, G. S. (2001) Kazhdan–-Lusztig polynomials for 321-hexagon-avoiding permutations. J. Algebraic Combin. 13 111–136.Google Scholar
[10]Bose, P., Buss, J. F. and Lubiw, A. (1998) Pattern matching for permutations. Inform. Process. Lett. 65 277–283.CrossRefGoogle Scholar
[11]Bousquet-Mélou, M. (2003) Four classes of pattern-avoiding permutations under one roof: Generating trees with two labels. Electron. J. Combin. 9 # 19 (electronic).Google Scholar
[12]Brignall, R., Ruškuc, N. and Vatter, V. Simple permutations: Decidability and unavoidable substructures. arXiv:math.CO/0609211.Google Scholar
[13]Brlek, S., Duchi, E., Pergola, E. and Rinaldi, S. (2005) On the equivalence problem for succession rules. Discrete Math. 298 142–154.Google Scholar
[14]Chow, T. and West, J. (1999) Forbidden subsequences and Chebyshev polynomials. Discrete Math. 204 119–128.Google Scholar
[15]Chung, F. R. K., Graham, R. L., Hoggatt, V. E. Jr. and Kleiman, M. (1978) The number of Baxter permutations. J. Combin. Theory Ser. A 24 382–394.Google Scholar
[16]Corneil, D. G., Lerchs, H. and Burlingham, L. S. (1981) Complement reducible graphs. Discrete Appl. Math. 3 163–174.Google Scholar
[17]Elder, M. and Vatter, V. (2005) Problems and conjectures presented at the Third International Conference on Permutation Patterns, University of Florida, March 7–11. arXiv:math.CO/0505504.Google Scholar
[18]Erdős, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Compos. Math. 2 463–470.Google Scholar
[19]Gessel, I. M. (1990) Symmetric functions and P-recursiveness. J. Combin. Theory Ser. A 53 257–285.CrossRefGoogle Scholar
[20]Green, R. M. and Losonczy, J. (2002) Freely braided elements of Coxeter groups. Ann. Comb. 6 337–348.CrossRefGoogle Scholar
[21]Knuth, D. E. (1969) The Art of Computer Programming, Vol. 1: Fundamental Algorithms, Addison-Wesley, Reading, MA.Google Scholar
[22]Kremer, D. (2000) Permutations with forbidden subsequences and a generalized Schröder number. Discrete Math. 218 121–130.Google Scholar
[23]Kremer, D. and Shiu, W. C. (2003) Finite transition matrices for permutations avoiding pairs of length four patterns. Discrete Math. 268 171–183.Google Scholar
[24]Le, I. (2005) Wilf classes of pairs of permutations of length 4. Electron. J. Combin. 12 # 25 (electronic).Google Scholar
[25]MacMahon, P. A. (1915/16) Combinatory Analysis, Cambridge University Press, London.Google Scholar
[26]Mansour, T. (2000) Permutations containing and avoiding certain patterns. In Formal Power Series and Algebraic Combinatorics (Moscow 2000), Springer, Berlin, pp. 704–708.Google Scholar
[27]Mansour, T. (2004) On an open problem of Green and Losonczy: Exact enumeration of freely braided permutations. Discrete Math. Theor. Comput. Sci. 6 461–470.Google Scholar
[28]Mansour, T. and Stankova, Z. (2003) 321-polygon-avoiding permutations and Chebyshev polynomials. Electron. J. Combin. 9 # 5 (electronic).Google Scholar
[29]Möhring, R. H. and Radermacher, F. J. (1984) Substitution decomposition for discrete structures and connections with combinatorial optimization. In Algebraic and Combinatorial Methods in Operations Research, Vol. 95 of North-Holland Math. Stud., North-Holland, Amsterdam, pp. 257–355.Google Scholar
[30]Nash-Williams, C. S. J. A. (1963) On well-quasi-ordering finite trees. Proc. Cambridge Philos. Soc. 59 833–835.CrossRefGoogle Scholar
[31]Sloane, N. J. A. The On-line Encyclopedia of Integer Sequences. Available online at: http://www.research.att.com/~njas/sequences/.Google Scholar
[32]Stankova, Z. E. (1994) Forbidden subsequences. Discrete Math. 132 291–316.CrossRefGoogle Scholar
[33]Stankova, Z. and West, J. (2004) Explicit enumeration of 321, hexagon-avoiding permutations. Discrete Math. 280 165–189.Google Scholar
[34]Stanley, R. P. (1997) Enumerative Combinatorics 1, Vol. 49 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge.Google Scholar
[35]Stanley, R. P. (1999) Enumerative Combinatorics 2, Vol. 62 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge.Google Scholar
[36]Vatter, V. (2006) Finitely labeled generating trees and restricted permutations. J. Symbolic Comput. 41 559–572.Google Scholar
[37]West, J. (1995) Generating trees and the Catalan and Schröder numbers. Discrete Math. 146 247–262.Google Scholar
[38]West, J. (1996) Generating trees and forbidden subsequences. Discrete Math. 157 363–374.Google Scholar
[39]Wilf, H. S. (1982) What is an answer? Amer. Math. Monthly 89 289–292.Google Scholar
[40]Zeilberger, D. (1998) Enumeration schemes and, more importantly, their automatic generation. Ann. Comb. 2 185–195.Google Scholar