Enumerations of specific permutation classes

In the study of permutation patterns, there has been considerable interest in enumerating specific permutation classes, especially those with relatively few basis elements. This area of study has turned up unexpected instances of Wilf equivalence, where two seemingly-unrelated permutation classes have the same number of permutations of each length.

Classes avoiding one pattern of length 3

There are two symmetry classes and a single Wilf class for single permutations of length three.

Classes avoiding one pattern of length 4

There are seven symmetry classes and three Wilf classes for single permutations of length four.

No non-recursive formula counting 1324-avoiding permutations is known. A recursive formula was given by Marinov & Radoičić (2003). A more efficient algorithm using functional equations was given by Johansson & Nakamura (2014), which was enhanced by Conway & Guttmann (2015), and then further enhanced by Conway, Guttmann & Zinn-Justin (2018) who give the first 50 terms of the enumeration. Bevan et al. (2020) currently have the best rigorously established lower and upper bounds for the growth rate of this class, having established that this growth rate lies in the interval [10.271, 13.5].

Classes avoiding two patterns of length 3

There are five symmetry classes and three Wilf classes, all of which were enumerated in Simion & Schmidt (1985).

Classes avoiding one pattern of length 3 and one of length 4

There are eighteen symmetry classes and nine Wilf classes, all of which have been enumerated. For these results, see Atkinson (1999) or West (1996).

Classes avoiding two patterns of length 4

There are 56 symmetry classes and 38 Wilf equivalence classes. Only 3 of these remain unenumerated, and their generating functions are conjectured not to satisfy any algebraic differential equation (ADE) by Albert et al. (2018); in particular, their conjecture would imply that these generating functions are not D-finite.

Heatmaps of each of the non-finite classes are shown on the right, from Albert et al. (2024). The lexicographically minimal symmetry is used for each class, and the classes are ordered in lexicographical order. To create each heatmap, one million permutations of length 300 were sampled uniformly at random from the class. The color of the point ( i , j ) {\displaystyle (i,j)} represents how many permutations have value j {\displaystyle j} at index i {\displaystyle i}. Higher resolution versions can be obtained at

See also

• Baxter permutation
• Riffle shuffle permutation

• Albert, Michael H.; Elder, Murray; Rechnitzer, Andrew; Westcott, P.; Zabrocki, Mike (2006), "On the Stanley–Wilf limit of 4231-avoiding permutations and a conjecture of Arratia", Advances in Applied Mathematics, 36 (2): 96–105, arXiv:, doi:, hdl:, MR.
• Albert, Michael H.; Atkinson, M. D.; Brignall, Robert (2011), (PDF), Pure Mathematics and Applications, 22: 87–98, arXiv:, MR.
• Albert, Michael H.; Atkinson, M. D.; Brignall, Robert (2012), , Electronic Journal of Combinatorics, 19 (3): Paper 20, 34 pp, doi:, MR.
• Albert, Michael H.; Atkinson, M. D.; Vatter, Vincent (2009), , Electronic Journal of Combinatorics, 16 (1): Paper 136, 9 pp, arXiv:, doi:, MR.
• Albert, Michael H.; Atkinson, M. D.; Vatter, Vincent (2014), (PDF), Australasian Journal of Combinatorics, 58 (1): 27–47, MR.
• Albert, Michael H.; Bean, Christian; Claesson, Anders; Nadeau, Émile; Pantone, Jay; Ulfarsson, Henning (2024), "Combinatorial Exploration: An algorithmic framework for enumeration", arXiv: [].
• Albert, Michael H.; Homberger, Cheyne; Pantone, Jay; Shar, Nathaniel; Vatter, Vincent (2018), "Generating permutations with restricted containers", Journal of Combinatorial Theory, Series A, 157: 205–232, arXiv:, doi:, MR.
• Atkinson, M. D. (1998), , Electronic Journal of Combinatorics, 5 R6: Paper 6, 13 pp, doi:, MR.
• Atkinson, M. D. (1999), "Restricted permutations", Discrete Mathematics, 195 (1–3): 27–38, doi:, MR.
• Atkinson, M. D.; Sagan, Bruce E.; Vatter, Vincent (2012), "Counting (3+1)-avoiding permutations", European Journal of Combinatorics, 33: 49–61, doi:, MR.
• Bevan, David (2015), "Permutations avoiding 1324 and patterns in Łukasiewicz paths", J. London Math. Soc., 92 (1): 105–122, arXiv:, doi:, MR.
• Bevan, David (2016a), (PDF), Australasian Journal of Combinatorics, 64 (1): 3–20, MR.
• Bevan, David (2016b), , Discrete Mathematics & Theoretical Computer Science, 18 (2) 1309: 14 pp, arXiv:, doi:.
• Bevan, David; Brignall, Robert; Elvey Price, Andrew; Pantone, Jay (2020), "A structural characterisation of Av(1324) and new bounds on its growth rate", European Journal of Combinatorics, 88 103115: 1–29, arXiv:, doi:.
• Bloom, Jonathan; Vatter, Vincent (2016), (PDF), Australasian Journal of Combinatorics, 64 (1): 77–87, MR.
• Bóna, Miklós (1997), "Exact enumeration of 1342-avoiding permutations: a close link with labeled trees and planar maps", Journal of Combinatorial Theory, Series A, 80 (2): 257–272, arXiv:, doi:, MR.
• Bóna, Miklós (1998), , Electronic Journal of Combinatorics, 5 R31: Paper 31, 12 pp, doi:, MR.
• Bóna, Miklós (2015), "A new record for 1324-avoiding permutations", European Journal of Mathematics, 1 (1): 198–206, arXiv:, doi:, MR.
• Callan, David (2013a), "The number of {1243, 2134}-avoiding permutations", Discrete Mathematics & Theoretical Computer Science 5287, arXiv:, doi:.
• Callan, David (2013b), "Permutations avoiding 4321 and 3241 have an algebraic generating function", Discrete Mathematics & Theoretical Computer Science 5286, arXiv:, doi:.
• Conway, Andrew; Guttmann, Anthony (2015), "On 1324-avoiding permutations", Advances in Applied Mathematics, 64: 50–69, doi:, MR.
• Conway, Andrew; Guttmann, Anthony; Zinn-Justin, Paul (2018), "1324-avoiding permutations revisited", Advances in Applied Mathematics, 96: 312–333, arXiv:, doi:.
• Gessel, Ira M. (1990), "Symmetric functions and P-recursiveness", Journal of Combinatorial Theory, Series A, 53 (2): 257–285, doi:, MR.
• Johansson, Fredrik; Nakamura, Brian (2014), "Using functional equations to enumerate 1324-avoiding permutations", Advances in Applied Mathematics, 56: 20–34, arXiv:, doi:, MR.
• Knuth, Donald E. (1968), The Art Of Computer Programming Vol. 1, Boston: Addison-Wesley, ISBN 978-0-201-89683-1, MR , OCLC.
• Kremer, Darla (2000), "Permutations with forbidden subsequences and a generalized Schröder number", Discrete Mathematics, 218 (1–3): 121–130, doi:, MR.
• Kremer, Darla (2003), "Postscript: "Permutations with forbidden subsequences and a generalized Schröder number"", Discrete Mathematics, 270 (1–3): 333–334, doi:, MR.
• Kremer, Darla; Shiu, Wai Chee (2003), "Finite transition matrices for permutations avoiding pairs of length four patterns", Discrete Mathematics, 268 (1–3): 171–183, doi:, MR.
• Le, Ian (2005), , Electronic Journal of Combinatorics, 12 R25: Paper 25, 27 pp, doi:, MR.
• MacMahon, Percy A. (1916), Combinatory Analysis, London: Cambridge University Press, MR.
• Marinov, Darko; Radoičić, Radoš (2003), , Electronic Journal of Combinatorics, 9 (2): Paper 13, 9 pp, doi:, MR.
• Miner, Sam (2016), "Enumeration of several two-by-four classes", arXiv: [].
• Miner, Sam; Pantone, Jay (2018), "Completing the structural analysis of the 2x4 permutation classes", arXiv: [].
• Pantone, Jay (2017), "The enumeration of permutations avoiding 3124 and 4312", Annals of Combinatorics, 21 (2): 293–315, arXiv:, doi:.
• Simion, Rodica; Schmidt, Frank W. (1985), "Restricted permutations", European Journal of Combinatorics, 6 (4): 383–406, doi:, MR.
• Vatter, Vincent (2006), "Finitely labeled generating trees and restricted permutations", Journal of Symbolic Computation, 41 (5): 559–572, arXiv:, doi:, MR.
• Vatter, Vincent (2012), "Finding regular insertion encodings for permutation classes", Journal of Symbolic Computation, 47 (3): 259–265, arXiv:, doi:, MR.
• West, Julian (1996), "Generating trees and forbidden subsequences", Discrete Mathematics, 157 (1–3): 363–374, doi:, MR.

External links

The , maintained by Bridget Tenner, contains details of the enumeration of many other permutation classes with relatively few basis elements.