Problems and conjectures

Vincent Vatter

doi:10.1017/CBO9780511902499.017

Problems and conjectures

Published online by Cambridge University Press: 05 October 2010

Edited by

Nik Ruškuc and

Steve Linton: Affiliation:
University of St Andrews, Scotland
Nik Ruškuc: Affiliation:
University of St Andrews, Scotland
Vincent Vatter: Affiliation:
Dartmouth College, New Hampshire

Book contents

Get access

Summary

A very brief introduction to permutation patterns

We say a permutation π contains or involves the permutation σ if deleting some of the entries of π gives a permutation that is order isomorphic to σ, and we write σ ≤ π. For example, 534162 contains 321 (delete the values 4, 6, and 2). A permutation avoids a permutation if it does not contain it.

This notion of containment defines a partial order on the set of all finite permutations, and the downsets of this order are called permutation classes. For a set of permutations B define Av(B) to be the set of permutations that avoid all of the permutations in B. Clearly Av(B) is a permutation class for every set B, and conversely, every permutation class can be expressed in the form Av(B).

For the problems we need one more bit of notation. Given permutations π and σ of lengths m and n, respectively, their direct sum, π ⊕ σ, is the permutation of length m + n in which the first m entries are equal to π and the last n entries are order isomorphic to σ while their skew sum, π ⊖ σ, is the permutation of length m + n in which the first m entries are order isomorphic to π while the last n entries are equal to π.

Type: Chapter
Information: Permutation Patterns , pp. 339 - 345

DOI: https://doi.org/10.1017/CBO9780511902499.017 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] M. H., Albert, R. E. L., Aldred, M. D., Atkinson, H. P., van Ditmarsch, B. D., Handley, C. C., Handley, and J., Opatrny. Longest subsequences in permutations. Australas. J. Combin., 28:225–238, 2003.Google Scholar

[2] M. H., Albert, M. D., Atkinson, R., Brignall, N., Ruškuc, R., Smith, and J., West. Growth rates for subclasses of Av(321). arXiv:0903.1999 [math.CO].

[3] M. H., Albert, M., Elder, A., Rechnitzer, P., Westcott, and M., Zabrocki. On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia. Adv. in Appl. Math., 36(2):95–105, 2006.Google Scholar

[4] M. H., Albert and S., Linton. Growing at a perfect speed. Combin. Probab. Comput., 18:301–308, 2009.Google Scholar

[5] R., Arratia. On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern. Electron. J. Combin., 6:Note, N1, 4 pp., 1999.Google Scholar

[6] M. D., Atkinson, M. M., Murphy, and N., Ruškuc. Sorting with two ordered stacks in series. Theoret. Comput. Sci., 289(1):205–223, 2002.Google Scholar

[7] E., Babson and E., Steingrímsson. Generalized permutation patterns and a classification of the Mahonian statistics. Sém. Lothar. Combin., 44:Article B44b, 18 pp., 2000.Google Scholar

[8] E., Babson and J., West. The permutations 123p4 … pm and 321p4 … pm are Wilf-equivalent. Graphs Combin., 16(4):373–380, 2000.Google Scholar

[9] J., Backelin, J., West, and G., Xin. Wilf-equivalence for singleton classes. Adv. in Appl. Math., 38(2):133–148, 2007.Google Scholar

[10] M., Bóna. Exact enumeration of 1342-avoiding permutations: a close link with labeled trees and planar maps. J. Combin. Theory Ser. A, 80(2):257–272, 1997.Google Scholar

[11] M., Bóna. A survey of stack-sorting disciplines. Electron. J. Combin., 9(2):Article 1, 16 pp., 2003.Google Scholar

[12] M., Bóna. The limit of a Stanley-Wilf sequence is not always rational, and layered patterns beat monotone patterns. J. Combin. Theory Ser. A, 110(2):223–235, 2005.Google Scholar

[13] A., Burstein and W., Stromquist. Dumont permutations of the third kind. Extended abstract, FPSAC 2007.

[14] M., Elder and V., Vatter. Problems and conjectures presented at the Third International Conference on Permutation Patterns, University of Florida, March 7–11, 2005. arXiv:0505504 [math.CO].

[15] M. T., Hardarson. Avoidance of partially ordered generalized patterns of the form κ-σ-κ. arXiv:0805.1872v1 [math.CO].

[16] A., Marcus and G., Tardos. Excluded permutation matrices and the Stanley-Wilf conjecture. J. Combin. Theory Ser. A, 107(1):153–160, 2004.Google Scholar

[17] V. R., Pratt. Computing permutations with double-ended queues, parallel stacks and parallel queues. In STOC '73: Proceedings of the fifth annual ACM symposium on Theory of computing, pages 268–277, New York, NY, USA, 1973.Google Scholar

[18] N., Ray and J., West. Posets of matrices and permutations with forbidden subsequences. Ann. Comb., 7(1):55–88, 2003.Google Scholar

[19] A., Regev. Asymptotic values for degrees associated with strips of Young diagrams. Adv. in Math., 41(2):115–136, 1981.Google Scholar

[20] C., Schensted. Longest increasing and decreasing subsequences. Canad. J. Math., 13:179–191, 1961.Google Scholar

[21] Z., Stankova and J., West. A new class of Wilf-equivalent permutations. J. Algebraic Combin., 15(3):271–290, 2002.Google Scholar

[22] Z. E., Stankova. Forbidden subsequences. Discrete Math., 132(1-3):291–316, 1994.Google Scholar

[23] V., Vatter. Permutation classes of every growth rate above 2.48188. Mathematika, 56:182–192, 2010.Google Scholar

[24] J., West. Permutations with forbidden subsequences and stack-sortable permutations. PhD thesis, M.I.T., 1990.

[25] J., West. Sorting twice through a stack. Theoret. Comput. Sci., 117(1-2):303–313, 1993.Google Scholar

Book contents

Problems and conjectures

Summary

Access options

References

Save book to Kindle

Save book to Dropbox

Save book to Google Drive